**PREPRINT**

國立臺灣大學 數學系 預印本 Department of Mathematics, National Taiwan University

### ~ mathlib/preprint/2012- 09.pdf

## On mixed domination problem in graphs

### James K. Lan and Gerard Jennhwa Chang

### September 24, 2012

### On mixed domination problem in graphs

^{I}

James K. Lan^{a,∗}, Gerard Jennhwa Chang^{a,b,c}

*a**Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan*

*b**Taida Institute for Mathematical Sciences, National Taiwan U., Taipei 10617, Taiwan*

*c**National Center for Theoretical Sciences, Taipei Oﬃce, Taiwan*

**Abstract**

*A mixed dominating set of a simple graph G = (V, E) is a subset D⊆ V ∪ E*
*such that every vertex or edge not in D is adjacent or incident to at least*
*one vertex or edge in D. The mixed domination problem is to determine a*
*minimum mixed dominating set of G. This paper studies mixed domination*
in graphs from an algorithmic point of view. In particular, a linear-time
labeling algorithm for the mixed domination problem in cacti is presented.

In addition, we ﬁx an incomplete proof of the NP-completeness of the mixed domination problem in split graphs [22]. Finally, we establish a primal-dual algorithm for the mixed domination problem in trees.

*Keywords:* Mixed domination, Cactus, Tree, Algorithm, NP-complete

**1. Introduction**

All graphs in this paper are simple, i.e., ﬁnite, undirected, loopless and
without multiple edges. Domination is a core NP-complete problem in graph
theory and combinatorial optimization. It has many applications in the real
world such as location problems, sets of representatives, social network the-
*ory, etc; see [14] for more interesting applications. A dominating set of a*
*graph G = (V, E) is a subset D of V such that every vertex not in D is*
*adjacent to at least one vertex in D. The domination number γ(G) of G is*

IThis research was partially supported by the National Science Council of the Republic of China under grants NSC100-2811-M-002-146 and NSC98-2115-M-002-013-MY3.

*∗*Corresponding author.

*Email addresses: drjamesblue@gmail.com (James K. Lan),*
gjchang@math.ntu.edu.tw (Gerard Jennhwa Chang)

*the minimum size of a dominating set of G. The domination problem is to*
*ﬁnd a minimum dominating set of G.*

The notion of dominating (or covering, interchangeably) vertices or edges
by other vertices or edges has been widely studied in the literature. Tra-
ditional (vertex) domination problem asks for dominating vertices by other
*vertices. Covering edges by vertices leads to the vertex cover problem. Cov-*
*ering vertices by edges results in the edge cover problem. When edges are to*
*be dominated by edges, we obtain the edge domination problem.*

The extension of the above notion is naturally considered. Namely, dom-
inating vertices and edges by other vertices and edges is studied [3, 4, 19,
*20, 22]. Speciﬁcally, given a graph G = (V, E), a vertex is said to mixed*
*dominate itself, its neighbors and all edges incident to it; an edge uv is said*
*to mixed dominate u and v and all edges incident to u or v. A mixed domi-*
*nating set of G is a subset D* *⊆ V ∪ E such that every element in V ∪ E is*
*mixed dominated by some element in D. In other words, every vertex and*
*every edge not in D is adjacent or incident to at least one element in D.*

*The mixed domination number γ*_{m}*(G) of G is the minimum size of a mixed*
*dominating set of G. The mixed domination problem is to ﬁnd a minimum*
*mixed dominating set of G.*

Mixed domination was introduced in early papers, and a mixed dominat-
*ing set is called a total cover in [3, 4, 19]. In this paper, we choose to follow*
the terminology of [14, 22]. Note that the meaning of mixed dominating set
(used in this paper and [14, 22]) is distinct from that in papers or books such
as [1, 8]. A practical application of mixed domination was introduced in [22]

as placing phase measurement units (PMUs) at selected vertices or edges to monitor theirs mixed neighbors’ state variables in an electric power system.

The domination problem is NP-complete for general graphs [13], and remains NP-complete for restricted classes of graphs such as bipartite graphs [6, 12], comparability graphs [12], chordal graphs [7, 9], planar graphs [13]

and split graphs [6]. Eﬃcient algorithms have been found in trees and interval graphs [8]. On the other hand, it has been proved that the mixed domination problem is NP-complete for general graphs [18], planar bipartite graphs [19]

and chordal graphs [16]. However, little work has been published on the
existence of eﬃcient algorithms to this problem. Majumdar [18] gave the ﬁrst
linear-time algorithm for the mixed domination problem in trees. Recently,
*Zhao et al. [22] gave a linear-time labeling algorithm for trees, and showed*
that the mixed domination problem remains NP-complete for split graphs.

*Unfortunately, we found that there is a ﬂaw in the algorithm of Zhao et al.*

and the NP-completeness for split graphs is incomplete. In addition, more eﬃcient algorithms for the mixed domination problem are still unknown and desired.

*The total graph T (G) of a graph G has V (G)∪ E(G) as its vertex set and*
*two vertices of T (G) are adjacent if and only if they are adjacent or incident*
*in G. It is clear that the mixed domination number of a graph is equal*
to the domination number of its total graph. Thus, one way to solve the
*mixed domination problem in G is to solve the domination problem in T (G).*

However, few graph classes are known to admit eﬃcient algorithms for the
domination problem. Even if a new graph class *F is found, it is still diﬃcult*
to ﬁnd graph class whose total graph is *F. Moreover, the transformation*
between the graph and its total graph may increase the time complexity of
the algorithm.

In this paper, we explore eﬃcient algorithms for the mixed domination problem in graphs. In Section 3, a linear-time labeling algorithm for the mixed domination problem in cacti is presented. A cactus can be viewed as an extension of trees. In Section 4, we point out the proof of the NP- completeness for the mixed domination problem in split graphs in [22] is incomplete, and provide a new proof to this result. In Section 5, a linear- time labeling algorithm based on the primal-dual approach for the mixed domination problem in trees is introduced. The presented algorithm can serve as an alternative solution to the mixed domination problem in trees.

**2. Definitions**

*Let G = (V, E) be a graph with vertex set V and edge set E. The*
*degree deg(v) of a vertex v in G is the number of incident edges to v. An*
*isolated vertex is a vertex v with deg(v) = 0. Deﬁne ε* *∈ V ∪ E to be*
*an element in G. For a vertex v in G, denote the open neighborhood by*
*N (v) =* *{ u ∈ V : uv ∈ E } and the closed neighborhood by N[v] = N(v) ∪*
*{ v }. Let E(v) = { e ∈ E : v ∈ e } and E[v] = { v } ∪ E(v). The closed mixed*
*neighborhood of an element ε in G is denoted by*

*N*_{m}*[ε] =*

{ *N (ε)∪ E[ε], if ε ∈ V ;*
*E[u]∪ E[v], if ε = uv ∈ E.*

*Then for elements ε*1 *and ε*2 *in G, ε*1 *is mixed dominated by ε*2 if and only if
*ε*_{1} *∈ N**m**[ε*_{2}*], and they are mixed neighbors of each other. A set D is a mixed*
*dominating set of G if and only if D∩ N**m**[ε]̸= ∅ for every element ε in G.*

*The subgraph of G induced by S* *⊆ V is the graph G[S] with vertex set*
*S and edge set* *{uv ∈ E : u, v ∈ S}. In a graph G = (V, E), the deletion of*
*S* *⊆ V from G, denoted by G − S, is the graph G[V \ S]; the deletion of*
*F* *⊆ E from G, denoted by G − F , is the graph (V, E \ F ). For an element ε*
*in G, we write G− ε for G − { ε }. The union of two graphs G*1 *and G*_{2} is the
*graph G*1*∪ G*2 *with vertex set V (G*1)*∪ V (G*2*) and edge set E(G*1)*∪ E(G*2).

*The length of a path is the number of edges in the path. The distance d(u, v)*
*from vertex u to vertex v in G is the minimum length of a path from u to v;*

*d(u, v) =* *∞ if there is no path from u to v.*

*A forest is a graph without cycles. A tree is a connected forest. A pendant*
*vertex in a graph is a vertex with degree one (the pendant vertex is also called*
*a leaf in a tree), and a pendant edge is an edge incident to a pendant vertex.*

*A penultimate vertex is a vertex all of whose neighbors except possibly one*
*are pendant vertices. A path of order n is denoted by P*_{n}*; a cycle of order n*
*is denoted by C**n*.

*In a graph G = (V, E), a stable set is a set of pairwise nonadjacent vertices*
*and a clique is a set of pairwise adjacent vertices. A vertex v is a cut-vertex*
*if the number of connected components is increased after removing v. A*
*block of a graph is a maximal connected subgraph without any cut-vertex.*

*An end-block of a graph is a block containing at most one cut-vertex of the*
*graph. A block graph is a graph whose blocks are cliques. A cactus is a*
connected graph whose blocks are either an edge or a cycle. Alternatively,
a cactus is a connected graph in which two cycles have at most one vertex
(cut-vertex) in common. A cactus is a tree if all the blocks are edges.

**3. Linear-time labeling algorithm for mixed domination in graphs**
Labeling techniques are widely used in the literatures for solving the
domination problem and its variants [5, 9, 10, 11, 15, 17, 21, 22]. Among
*them, the FBR labeling technique solves a slightly more general domination*
*problem, which can be formulated as follows. For a graph G = (V, E), V*
*is partitioned into three disjoint sets F, B and R, where F, B and R are*
*called free set, bound set and required set, respectively. An optional domi-*
*nating set of G (with respect to F, B and R) is a subset D* *⊆ V such that*
*R* *⊆ D and D dominates B. Note that optional domination is the ordinary*
*domination when specifying B = V and F = R =∅.*

Based on this technique, Cockayne, Goodman and Hedetniemi [11] gave
*the ﬁrst linear-time algorithm for the domination problem in trees. Laskar et*

*al. [17] further partitioned V into F* *·∪B ·∪R*1*·∪R*2 *to ﬁnd a subset D⊆ V such*
*that R*_{1} *∪ R*2 *⊆ D and dominates B ∪ R*1 *for the total domination problem*
*in trees. Zhao et al. [22] gave a linear-time labeling algorithm for the mixed*
*domination problem in trees by partitioning V* *∪ E = F ·∪B ·∪R, specifying*
*B = V* *∪ E, F = R = ∅ and replacing domination with mixed domination.*

In this section, we employ the FBR labeling technique to ﬁnd a minimum
mixed dominating set in a cactus. Unlike the labeling method used in [22],
*V* *∪ E is partitioned into four disjoint sets F, B, B*^{′}*and R in our algorithm,*
*where B*^{′}*is called strictly bound set. More speciﬁcally, given a graph G =*
*(V, E), an FBR assignment L is a mapping that assigns each element ε in G*
a label^{1}

*L(ε)∈*

{ *{ F, B, B*^{′}*, R} , if ε ∈ V ;*
*{ F, B, R } ,* *if ε∈ E.*

*Namely, any element in G can be assigned a label in* *{ F, B, B*^{′}*, R}, except*
*that only vertices can be assigned a label B*^{′}*. Thus, V* *∪ E = F ·∪B ·∪B*^{′}*·∪R.*

*An FBR mixed dominating set of G (with respect to L) is a subset D* *⊆ V ∪E*
*such that R⊆ D and*

*• for every element ε ∈ B, D ∩ N**m**[ε]̸= ∅, and*

*• for every vertex v ∈ B*^{′}*, D∩ E[v] ̸= ∅.*

*That is, D contains all required elements, and every bound element not in D*
*is adjacent or incident to at least one element in D, and every strictly bound*
*vertex not in D is incident to at least one edge in D. Free elements need not*
*be mixed dominated, but can be included in D to mixed dominate elements*
*in B∪B*^{′}*. The FBR mixed domination number γ*_{L}*(G) is the minimum size of*
*an FBR mixed dominating set in G; such set is called a γ*L*-set of G. Notice*
*that if we specify B** ^{′}* =

*∅, then FBR mixed domination is the optional mixed*

*domination used in [22]; if we specify that B = V*

*∪ E and F = B*

^{′}*= R =∅,*

*then γ*L

*(G) = γ*

*m*

*(G). Thus an algorithm for computing the value of γ*L

*(G)*

*is suﬃcient to compute the value of γ*

_{m}*(G).*

*Denote R = B* *∪ B*^{′}*. Let G be a graph with an FBR assignment L.*

*Suppose ε is an element in G and ε* *∈ R. It is clear that ε must be included in*
*any FBR mixed dominating set of G. Hence, for every element ε*^{′}*∈ N**m**[ε]∩B*
*we have D∩ N**m**[ε** ^{′}*]

*̸= ∅, where D is a minimum FBR mixed dominating set*

1*For simplicity, the terms F, B, B*^{′}*, R represent sets and labels interchangeably.*

*of G. Moreover, if ε is an edge, then for every vertex v* *∈ N**m**[ε]∩ B** ^{′}* we have

*D∩ E[v] ̸= ∅. Therefore these elements no longer need to retain its label.*

*Release is a procedure to relabel these elements with F .*

**Procedure Release(ε)**

1 **if ε is a vertex then**

2 *L(ε** ^{′}*)

*← F for each ε*

^{′}*∈ N*

*m*

*[ε]∩ B;*

3 **else // ε is an edge**

4 *L(ε** ^{′}*)

*← F for each ε*

^{′}*∈ N*

*m*

*[ε]∩ R;*

*3.1. Finding a γ*_{L}*-set in a tree and a cycle*

In this subsection, we present algorithms to ﬁnd a minimum FBR mixed dominating set in trees and cycles. The presented algorithm will be used for ﬁnding a minimum FBR mixed dominating set in a cactus.

**Theorem 3.1. Suppose G is a graph with an FBR assignment L. Suppose***C is an end-block of G and* *|C| = 2. Let x be the unique cut-vertex of C and*
*let y be the pendant vertex of C. Let G*^{′}*= G− y and let G*^{′}*c* *denote the graph*
*which results from G*^{′}*by relabeling x with c∈ { F, B, B*^{′}*, R}. Let L*^{′}*and L*^{′′}

*be those restrictions of L on G*^{′}*and G*^{′′}*, respectively.*

*(1) If y* *∈ F and xy ∈ F , then γ*L*(G) = γ*_{L}*′**(G*^{′}*).*

*(2) If y* *∈ F and xy ∈ B, then γ*L*(G) = γ*_{L}*′**(G*^{′}_{B}*′**).*

*(3) If y* *∈ B and xy ̸∈ R, then γ*L*(G) = γ*_{L}*′**(G*^{′}_{R}*).*

*(4) If y* *∈ B*^{′}*and xy* *̸∈ R, then γ*L*(G) = γ*L^{′′}*(G*^{′′}*), where G*^{′′}*results from G*
*by relabeling y and xy with F and R, respectively.*

*(5) Suppose y* *∈ R or xy ∈ R. Let S = { y, xy } ∩ R. Then γ*L*(G) =*
*γ*_{L}*′′**(G** ^{′′}*) +

*|S|, where G*

^{′′}*results from G by releasing ε for each ε∈ S and*

*then deleting y.*

*Proof. We shall only provide the proofs of cases (2), (4) and (5); other cases*
are similar to obtain.

*(2) Suppose D*^{′}*is a γ*_{L}*′**-set of G*^{′}_{B}_{′}*. Since x∈ B** ^{′}*, there exists an element

*ε∈ E[x] in G*

^{′}

_{B}*′*

*such that ε∈ D*

^{′}*. Then D*

*is also an FBR mixed dominating*

^{′}*set of G, since y is free and xy is mixed dominated by ε. Thus γ*L

^{′}*(G*

^{′}

_{B}*′*)

*≥*

*γ*

_{L}

*(G). Conversely, suppose D is a γ*

_{L}

*-set of G. By the minimality of D, x, y*

*and xy cannot concurrently be included in D. Let ε be y or xy. If ε* *∈ D,*
*then D*^{′}*= D− { ε } ∪ { x } is also an FBR mixed dominating set of G*^{′}_{B}*′* and

*|D*^{′}*| = |D|. Thus γ*L*(G)* *≥ γ*L^{′}*(G*^{′}_{B}*′*).

*(4) Suppose D*^{′}*is a γ*_{L}*′′**-set of G*^{′′}*. Since xy* *∈ R in G*^{′′}*, xy∈ D*^{′}*. Then D*^{′}*is an FBR mixed dominating set of G, since D∩ E[y] ̸= ∅. Thus γ*L^{′′}*(G** ^{′′}*)

*≥*

*γ*L

*(G). Conversely, suppose D is a γ*L

*-set of G. Since y*

*∈ B*

^{′}*, D∩ E[y] =*

*D∩ { y, xy } ̸= ∅. By the minimality of D, y and xy cannot be both included*

*in D. If y*

*∈ D, then D*

^{′}*= D−{ y }∪{ xy } is also an FBR mixed dominating*

*set of G*

*and*

^{′′}*|D*

^{′}*| = |D|. Thus, γ*L

*(G)≥ γ*L

^{′′}*(G*

*).*

^{′′}*(5) Suppose D*^{′}*is a γ*_{L}*′′**-set of G*^{′′}*. Clearly D*^{′}*∪ S is an FBR mixed*
*dominating set of G. Thus γ*_{L}*′′**(G** ^{′′}*) +

*|S| ≥ γ*L

*(G). Conversely, suppose D is*

*a γ*L

*-set of G. For each ε∈ S, we have ε ∈ D. We only show the case y ∈ R*

*and xy*

*̸∈ R; other cases are similar to obtain. Clearly, we have |S| = 1 and*

*y*

*∈ D. Since xy ̸∈ R, by the minimality of D, x and xy cannot be both in*

*D. If x*

*∈ D, then D − { y } is clearly an FBR mixed dominating set of G*

*.*

^{′′}*If xy*

*∈ D, then D − { y, xy } ∪ { x } is an FBR mixed dominating set of G*

*.*

^{′′}*If x*

*̸∈ D and xy ̸∈ D, then since all the elements in N*

*m*

*[y]∩ B have been*

*relabeled with F in G*

^{′′}*, D− { y } is an FBR mixed dominating set of G*

*. In*

^{′′}*either case, γ*

_{L}

*(G)− |S| ≥ γ*L

^{′′}*(G*

*).*

^{′′}Based on the above theorem, we design a procedure, called MixDomLeaf, to handle the case of an end-block with cardinality two. The procedure will be used in algorithms for the FBR mixed domination problem in trees and cacti.

**Procedure MixDomLeaf(G, C)**

1 *let x be the unique cut-vertex of C and y be its neighbor;*

2 **if L(y) = R or L(xy) = R then**

3 *S* *← { ε: ε ∈ { y, xy } , ε ∈ R };*

4 *D← D ∪ S;*

5 **foreach ε****∈ S do Release(ε);**

6 **else**

7 **if L(y) = B then L(x)**← R;

8 **if L(y) = B**^{′}**then D***← D ∪ { xy } and Release(xy);*

9 **if L(y) = F and L(xy) = B then L(x)**← B* ^{′}*;

10 *G← G − y;*

Next, we present an algorithm, called MixDomTree to ﬁnd a minimum FBR mixed dominating set in a tree.

**Algorithm 1. MixDomTree (Finding a γ**_{L}-set of a tree)
**Input: A tree T = (V, E) with an FBR assignment L.**

**Output: A minimum FBR mixed dominating set D of T .**

1 *T*^{′}*← T ;*

2 *D← ∅;*

3 **while** *|T*^{′}**| > 1 do**

4 *let C be an end-block of T** ^{′}*;

5 *MixDomLeaf(T*^{′}*, C);*

6 **end**

7 *let the only left vertex of T*^{′}*be x;*

8 **if x****̸∈ F then D ← D ∪ { x };**

**Theorem 3.2. Algorithm MixDomTree finds a minimum FBR mixed domi-***nating set in a tree in linear-time.*

*Proof. The correctness comes from Theorem 3.1 and the fact that every*
induced subgraph of a tree of size greater than one has an end-block with
cardinality two. Besides, MixDomTree runs in linear-time, since it visits each
vertex once, and all of the statements within which can be executed in at
*most O(deg(x)) time, where x is the neighbor of a pendant vertex y.*

**Remark. Algorithm MixDomTree is a one stage algorithm and it can also**
ﬁnd a minimum FBR mixed dominating set of a forest without modifying the
algorithm. We notice that the labeling algorithm for the mixed domination
problem in trees in [22] is a two-stage algorithm and it has a ﬂaw shown as
*follows. Let T be a rooted tree with V (T ) =* *{ x, y, z }, E(T ) = { xy, xz }*
*and root=x. Let all the elements of T be labeled F except that y is labeled*
*R and xz is labeled B. The algorithm will execute Stage B, return D =*
*{ y } and then stop, where D is the output of the algorithm. However, D*
*is not a minimum optional mixed dominating set because xz is not mixed*
*dominated by any element in D. To ﬁx this ﬂaw, modify Stage B as follows:*

*D← D ∪ R. If there exists an element ε ∈ B such that D ∩ N**m**[ε] =∅, then*
*L(root) = R, D* *← D ∪ {root} and stop; Otherwise, stop. See [22] for further*
details.

Now we consider ﬁnding a minimum FBR mixed dominating set of a cycle. The idea is to cut the cycle into a path and then use MixDomTree.

*Let ε be an element in C. Denote the cycle C with element ε relabeled with*
*c∈ { F, B, B*^{′}*, R} by C**ε** ^{c}*.

**Algorithm 2. MixDomCYC (Finding a γ**_{L}-set of a cycle)
**Input: A cycle C with an FBR assignment L.**

**Output: A minimum FBR mixed dominating set D of C.**

1 *D← ∅;*

2 **if** **∃ an element ε ∈ R then**

3 *Release(ε) and D* *← MixDomTree(C − ε) ∪ { ε };*

4 **else**

5 **if** *∃ an element ε ∈ B or B*^{′}**then**

6 **if ε****∈ B then S ← N***m***[ε] else S***← E[ε];*

7 *U*_{min}*← ∞;*

8 **foreach ε***i* **in S do**

9 *U*_{i}*← MixDomCYC(C**ε*^{R}*i*);

10 **if** *|U**i**| < |U**min***| then min ← i;**

11 *D← U**min*;

Based on Theorem 3.2, the construction and correctness of MixDomCYC is straightforward, and the proof is omitted. For the time complexity, since Mix- DomTree is linear and MixDomCYC makes at most ﬁve calls to MixDomTree (the number of mixed neighbors of an element in a cycle), it is clear that MixDomCYC is linear.

*3.2. Finding a γ*_{L}*-set in cacti*

In this subsection, we present an algorithm to ﬁnd a minimum FBR
mixed dominating set in a cactus. The construction and correctness of the
*algorithm is based on the following theorem. Note that for a cycle C with*
*an FBR assignment L and a vertex v in C, γ*_{L}*′**(C*_{v}* ^{F}*)

*≤ γ*L

^{′}*(C*

_{v}*)*

^{B}*≤ γ*L

^{′}*(C*

_{v}

^{B}*)*

^{′}*≤*

*γ*L

^{′}*(C*

_{v}

^{R}*) < γ*L

^{′}*(C*

_{v}

^{F}*) + 1 holds, where L*

^{′}*is the same as L with the above*

*modiﬁcations on v.*

**Theorem 3.3. Suppose K is a cactus with an FBR assignment L. Suppose***C is an end-block of K and* *|C| ≥ 3. Let x be the unique cut-vertex of C and*
*y, z be the two neighbors of x on C. Let K*^{′}*denote the cactus which results*
*from K by deleting all vertices only in C. Let K*_{c}^{′}*denote the cactus which*
*results from K*^{′}*by relabeling x with c* *∈ { F, B, B*^{′}*, R}. Let L*^{′}*and L*^{′′}*be*
*those restrictions of L on K*^{′}*and C, respectively.*

*(1) If x∈ R, then γ*L*(K) = γ*_{L}*′**(K*^{′}*) + γ*_{L}*′′**(C)− 1.*

*(2) Suppose x̸∈ R and γ*L^{′′}*(C) < γ*_{L}*′′**(C*_{x}^{R}*).*

*(2.1) If γ*_{L}*′′**(C) < γ*_{L}*′′**(C*_{xy}^{R}*) and γ*_{L}*′′**(C) < γ*_{L}*′′**(C*_{xz}^{R}*), then γ*_{L}*(K) =*
*γ*_{L}*′**(K*_{F}^{′}*) + γ*_{L}*′′**(C).*

*(2.2) If γ*_{L}*′′**(C) = γ*_{L}*′′**(C*_{xy}^{R}*) or γ*_{L}*′′**(C) = γ*_{L}*′′**(C*_{xz}^{R}*), then γ*_{L}*(K) = γ*_{L}*′**(K** ^{′′}*)+

*γ*_{L}*′′**(C*_{e}^{R}*), where K*^{′′}*results from K by releasing e, where e = xy or*
*xz in which γ*_{L}*′′**(C) = γ*_{L}*′′**(C*_{e}^{R}*), and then deleting all vertices only*
*in C.*

*(3) Suppose x∈ B or B*^{′}*and γ*_{L}*′′**(C*_{x}^{F}*) < γ*_{L}*′′**(C) = γ*_{L}*′′**(C*_{x}^{R}*). Then γ*_{L}*(K) =*
*γ*_{L}*′**(K*^{′}*) + γ*_{L}*′′**(C*_{x}^{F}*).*

*(4) Suppose x̸∈ R and γ*L^{′′}*(C*_{x}^{F}*) = γ*_{L}*′′**(C) = γ*_{L}*′′**(C*_{x}^{R}*).*

*(4.1) If γ*L^{′′}*(C) < γ*L^{′′}*(C*_{xy}^{R}*) or γ*L^{′′}*(C) < γ*L^{′′}*(C*_{xz}^{R}*) or γ*L^{′′}*(C) = γ*L^{′′}*(C*_{xy}* ^{R}*) =

*γ*

_{L}

*′′*

*(C*

_{xz}

^{R}*) = γ*

_{L}

*′′*

*(C*

*− x) then γ*L

*(K) = γ*

_{L}

*′*

*(K*

_{R}

^{′}*) + γ*

_{L}

*′′*

*(C*

_{x}*)*

^{R}*− 1.*

*(4.2) If γ*_{L}*′′**(C) = γ*_{L}*′′**(C*_{xy}^{R}*) = γ*_{L}*′′**(C*_{xz}^{R}*) = γ*_{L}*′′**(C* *− x) + 1, then γ*L*(K) =*
*γ*L^{′}*(K*^{′′}*) + γ*L^{′′}*(C* *− x), where K*^{′′}*results from K by deleting all*
*elements in C\ N**m**[x], and then relabels y and z with F .*

*Proof. We shall only provide the proofs of cases (2) and (4); other cases are*
similar to obtain.

*(2) In this case, we give a proof for the case x∈ B; the argument can be*
*applied to cases x* *∈ F and x ∈ B*^{′}*. Note that γ*_{L}*′′**(C*_{x}^{F}*) = γ*_{L}*′′**(C) < γ*_{L}*′′**(C*_{x}* ^{R}*).

*(2.1) Let D*_{1} *and D*_{2} *be γ*_{L}*′**-set of K*_{F}^{′}*and γ*_{L}*′′**-set of C, respectively.*

*Clearly, D*_{1}*∪ D*2 *is an FBR mixed dominating set of K and thus γ*_{L}*′**(K*_{F}* ^{′}* ) +

*γ*

_{L}

*′′*

*(C)≥ γ*L

*(K). Conversely, suppose D is a γ*

_{L}

*-set of K. We have two cases.*

*Case 1 : x* *∈ D. Since γ*L^{′′}*(C) < γ*_{L}*′′**(C*_{x}* ^{R}*),

*|D ∩ C| > γ*L

^{′′}*(C). Let C*

^{′}*be a γ*

_{L}

*′′*

*-set of C and let D*

^{′}*= (D*

*− C) ∪ { x } ∪ C*

*. Then*

^{′}*|D| = |D*

^{′}*|*

*and D*

^{′}*is a FBR mixed dominating set of K. Clearly, D*

^{′}*− C*

^{′}*and C*

*are*

^{′}*FBR mixed dominating sets of K*

_{F}

^{′}*and C, respectively. We have γ*

_{L}

*(K)*

*≥*

*γ*

_{L}

*′*

*(K*

_{F}

^{′}*) + γ*

_{L}

*′′*

*(C).*

*Case 2 : x̸∈ D. Since x ∈ B, there exists an element ε ∈ D ∩ N**m**[x].*

*Subcase 2-1 : ε* *∈ C. Suppose ε = xy or xz. Since γ*L^{′′}*(C) < γ*_{L}*′′**(C*_{ε}* ^{R}*),

*|D ∩ C| > γ*L^{′′}*(C). Let C*^{′}*be a γ*_{L}*′′**-set of C. Then D*^{′}*= (D− C) ∪ { x } ∪ C*^{′}*is also a γ*_{L}*-set of K in which x* *∈ D** ^{′}*. Thus, we go back to Case 1 and

*γ*

_{L}

*(K)*

*≥ γ*L

^{′}*(K*

_{F}

^{′}*) + γ*

_{L}

*′′*

*(C) holds. Now suppose ε is a vertex in C. Then*

*clearly D*

*− C and D ∩ C are FBR mixed dominating sets of K*

*F*

^{′}*and C,*

*respectively. We have γ*

_{L}

*(K)≥ γ*L

^{′}*(K*

_{F}

^{′}*) + γ*

_{L}

*′′*

*(C).*

*Subcase 2-2 : ε̸∈ C. In this case, we have |D ∩ C| ≥ γ*L^{′′}*(C). Let C** ^{′}* be a

*γ*

_{L}

*′′*

*-set of C and let D*

^{′}*= (D− C) ∪ C*

^{′}*. Now we have a γ*

_{L}

*-set of K in which*

*x is mixed dominated by some element in C. As in the previous subcase,*
*γ*_{L}*(K)≥ γ*L^{′}*(K*_{F}^{′}*) + γ*_{L}*′′**(C).*

*(2.2) W.l.o.g. assume that e = xy, i.e., γ*L^{′′}*(C) = γ*L^{′′}*(C*_{xy}^{R}*). Let D*1 and
*D*_{2} *be γ*_{L}*′**-set of K*^{′′}*and γ*_{L}*′′**-set of C*_{xy}^{R}*, respectively. Then D*_{1}*∪D*2is an FBR
*mixed dominating set of K, since the elements that have been relabeled with*
*F in K*^{′′}*are mixed dominated by e. Hence γ*L^{′}*(K*^{′′}*) + γ*L^{′′}*(C*_{xy}* ^{R}*)

*≥ γ*L

*(K).*

*Conversely, suppose D is a γ*_{L}*-set of K. Since γ*_{L}*′′**(C) < γ*_{L}*′′**(C*_{x}* ^{R}*), we have

*|D ∩ C| ≥ γ*L^{′′}*(C). Let C*^{′}*be a γ*_{L}*′′**-set of C*_{xy}^{R}*, which is also a γ*_{L}*′′**-set of C.*

*Let D*^{′}*= (D− C) ∪ { x } ∪ C*^{′}*. Then D*^{′}*is also a γ*L*-set of K. D*^{′}*− C** ^{′}* and

*C*

^{′}*are FBR mixed dominating sets of K*

^{′′}*and C*

_{xy}*, respectively. We have*

^{R}*γ*

_{L}

*(K)≥ γ*L

^{′}*(K*

^{′′}*) + γ*

_{L}

*′′*

*(C*

_{xy}*).*

^{R}*(4.1) Let D*1 *and D*2 *be γ*L^{′}*-set of K*_{R}^{′}*and γ*L^{′′}*-set of C*_{x}* ^{R}*, respectively.

*Then x* *∈ D*1*∩ D*2*. Clearly D*_{1} *∪ D*2 *is an FBR mixed dominating set of K*
*and we have γ*_{L}*′**(K*_{R}^{′}*) + γ*_{L}*′′**(C*_{x}* ^{R}*)

*− 1 ≥ γ*L

*(K). Conversely, suppose D is a*

*γ*L

*-set of K. Two cases.*

*Case 1 : x∈ D. Then (D−C)∪{x} and D∩C are FBR mixed dominating*
*sets of K*_{R}^{′}*and C*_{x}^{R}*, respectively. We have γ*_{L}*(K)≥ γ*L^{′}*(K*_{R}^{′}*) + γ*_{L}*′′**(C*_{x}* ^{R}*)

*− 1.*

*Case 2 : x* *̸∈ D. We claim that |D ∩ C| ≥ γ*L^{′′}*(C). Suppose on the*
contrary that *|D ∩ C| = γ*L^{′′}*(C)− 1. Suppose γ*L^{′′}*(C) < γ*_{L}*′′**(C*_{e}* ^{R}*) for some

*e*

*∈ { xy, xz }. Then (D ∩ C) ∪ { e } results in a γ*L

^{′′}*-set of C, contradicts to*

*the assumption of γ*L

^{′′}*(C) < γ*L

^{′′}*(C*

_{e}

^{R}*). Now suppose γ*L

^{′′}*(C) = γ*L

^{′′}*(C*

_{xy}*) =*

^{R}*γ*

_{L}

*′′*

*(C*

_{xz}

^{R}*) = γ*

_{L}

*′′*

*(C−x). Note that |D ∩ C| = γ*L

^{′′}*(C)−1 implies γ*L

^{′′}*(C−x) ≤*

*γ*

_{L}

*′′*

*(C)− 1, contradicts to the assumption of γ*L

^{′′}*(C) = γ*

_{L}

*′′*

*(C*

*− x). Hence*

*we have the claim. Now let C*

^{′}*be a γ*L

^{′′}*-set of C*

_{x}

^{R}*, which is also a γ*L

*-set of*

^{′′}*C. Let D*

^{′}*= (D− C) ∪ C*

*. We have*

^{′}*|D| = |D*

^{′}*| and D*

^{′}*is also a γ*

_{L}

*-set of K*

*in which x∈ D*

*. By Case 1, we have the desired result.*

^{′}*(4.2) Let D*1 *and D*2 *be γ*L^{′}*-set of K*^{′′}*and γ*L^{′′}*-set of C* *− x, respectively.*

*Clearly D*_{1}*∪D*2 *is an FBR mixed dominating set of K and we have γ*_{L}*′**(K** ^{′′}*)+

*γ*_{L}*′′**(C* *− x) ≥ γ*L*(K). Conversely, suppose D is a γ*_{L}*-set of K. Two cases:*

*Case 1 :* *|D ∩ C| ≥ γ*L^{′′}*(C).* *Let D*^{′′}*be a γ*L^{′′}*-set of C* *− x, and let*
*D*^{′}*= (D* *− C) ∪ { x } ∪ D*^{′′}*. Since γ*_{L}*′′**(C) = γ*_{L}*′′**(C* *− x) + 1, |D| = |D*^{′}*|*
*and thus D*^{′}*is also a γ*_{L}*-set of K. Clearly, D*^{′}*∩ K*^{′′}*and D*^{′}*∩ (C − x) are*
*FBR mixed dominating sets of K*^{′′}*and C* *− x, respectively. Thus γ*L*(K)* *≥*
*γ*_{L}*′**(K*^{′′}*) + γ*_{L}*′′**(C* *− x).*

*Case 2 :* *|D ∩ C| = γ*L^{′′}*(C)− 1. In this case, we have x ̸∈ D, xy ̸∈ D, xz ̸∈*

*D, and at least one of x, xy and xz must be mixed dominated by an element*
*in D− C. We claim that there must exist an edge e such that e ∈ D − C and*
*x* *∈ e. Suppose the claim is not true. Then D ∩ C induces an FBR mixed*

*dominating set of C*_{x}^{F}*, contradicts to the assumption of γ*_{L}*′′**(C*_{x}^{F}*) = γ*_{L}*′′**(C).*

*As a result, D∩ K*^{′′}*and D∩ (C − x) are clearly FBR mixed dominating sets*
*of K*^{′′}*and C− x, respectively. Thus γ*L*(K)≥ γ*L^{′}*(K*^{′′}*) + γ*L^{′′}*(C− x).*

Based on the Theorem 3.3, we design a procedure, called MixDomC3, to handle the case of an end-block with cardinality larger than two.

**Procedure MixDomC3(G, C)**

1 *let x be the unique cut-vertex of C and y, z be its neighbors;*

/* Compute the following sets only when they are needed. */

*U* *← MixDomCYC(C), U**R**← MixDomCYC(C**x*^{R}*), U**F* *← MixDomCYC(C**x** ^{F}*),

*U*

*xy*

*← MixDomCYC(C*

*xy*

^{R}*), U*

*xz*

*← MixDomCYC(C*

*xz*

^{R}*), U*

*P*

*← MixDomTree(C − x);*

2 * if L(x) = R then D← D ∪ U;* /* Case 1 */

3 **else if** *|U| < |U**R***| then**

4 **if** *|U| < |U**xy**| and |U| < |U**xz** | then* /* Case 2.1 */

5 *L(x)← F and D ← D ∪ U;*

6 **else** /* Case 2.2 */

7 *choose e∈ { xy, xz } such that |U| = |U**e**|;*

8 *Release(e) and D* *← D ∪ U**e*;

9 **else**

10 **if L(x)**̸= F and |U*F** | < |U| then* /* Case 3 */

11 *D← D ∪ U**F*;

12 **else if** *|U| < |U**xy**| or |U| < |U**xz**| or |U| = |U**P**|* **then** /* Case 4.1 */

13 *L(x)← R and D ← D ∪ U**R*;

14 **else** /* Case 4.2 */

15 *L(y)← F , L(z) ← F , D ← D ∪ U**P* *and Case← 4.2;*

16 **if Case = 4.2 then**

17 *G← G − { ε ∈ C : ε ̸∈ N**m**[x]};*

18 **else**

19 *G← (G − C) ∪ { x };*

We are now ready to present our algorithm, called MixDomCAC, to de- termine a minimum FBR mixed dominating set in a cactus. Our algorithm takes MixDomTree and MixDomCYC as subroutines, which ﬁnds a minimum FBR mixed dominating sets of a tree and a cycle, respectively.

Figure 1: Mixed domination in a cactus.

**Algorithm 3. MixDomCAC (Finding a γ**_{L}-set of a cactus)
**Input: A cactus K = (V, E) with an FBR assignment L.**

**Output: A minimum FBR mixed dominating set D of K.**

1 *K*^{′}*← K;*

2 *D← ∅;*

3 **while** *|K*^{′}**| ̸= ∅ do**

4 **if K**^{′}**is a block then**

5 **if** *|K*^{′}**| < 3 then**

6 *D← D ∪ MixDomTree(K*^{′}**) and stop;**

7 **else**

8 *D← D ∪ MixDomCYC(K*^{′}**) and stop;**

9 *let C be an end-block of K** ^{′}*;

10 **if** **|C| = 2 then MixDomLeaf(K**^{′}**,C) else MixDomC3(K**^{′}*,C);*

11 **end**

It is well-known that cacti can be recognized in linear-time [2, 15]. For each end-block of a cactus, Algorithm MixDomCAC calls MixDomTree at most one time and calls MixDomCYC at most ﬁve times. Since MixDomTree and MixDomCYC are linear, algorithm MixDomCAC is clearly linear. Fig. 1 shows an example of a minimum mixed dominating set in a cactus.

**Theorem 3.4. Algorithm MixDomCAC finds a minimum FBR mixed domi-***nating set in a cactus in linear-time.*

**4. NP-completeness of mixed domination in split graphs**

In this section, we study the complexity of the mixed domination problem:

MIXED DOMINATION

*INSTANCE: A graph G = (V, E) and positive integers k.*

*QUESTION: Does G have a mixed dominating set of size* *≤ k?*

*In [22], Zhao et al. showed that the mixed domination problem remains*
*NP-complete for split graphs. A split graph is a graph whose vertex set*
is the disjoint union of a clique and a stable set. Unfortunately, the proof
in [22] is incomplete. In the following, we indicate the ﬂaw and present a
counterexample to their proof.

*To show the NP-completeness of MIXED DOMINATION, Zhao et al. [22]*

adopted the reduction from a well-known NP-complete problem, namely the
*vertex cover problem for general graphs. A vertex cover of a graph G = (V, E)*
*is a subset C* *⊆ V such that for every edge uv ∈ E we have u ∈ C or v ∈ C.*

*The vertex cover problem is to ﬁnd a minimum vertex cover of G.*

Let the decision version of the vertex cover problem be denoted by VER-
TEX COVER. The transformation is constructed as follows. Given an in-
*stance of VERTEX COVER, construct the graph G*^{′}*= (V*^{′}*, E*^{′}*) with V** ^{′}* =

*V*

*∪ E and E*

*=*

^{′}*{ uv : u ̸= v, u, v ∈ V } ∪ { ve: v ∈ V, e ∈ E, v ∈ e }. Then*showed that:

*G has a vertex cover with cardinality k if and only if the split*
*graph G*^{′}*has a mixed dominating set of size k +*⌈_{n}_{−k}

2

⌉. (*∗)*

For the suﬃciency part of (*∗), suppose the split graph G** ^{′}* has a mixed dom-

*inating set D of size k +*⌈

_{n−k}2

⌉*. Zhao et al. [22] assumed that D contains*
*no edge between V and E by claiming that if there exists some edge ve∈ D*
*such that v* *∈ V and e ∈ E in G*^{′}*, then edge ve can be replaced by v in D*
*without changing the size of D. However, the claim is not always true. Take*
*Fig. 2 for example. Suppose G*^{′}*has a mixed dominating set D =* *{ ae, cf, d }*
*of size three. It is clear that D** ^{′}* =

*{ a, c, d } is not a mixed dominating set of*

*G*

^{′}*, since the edge bf is not mixed dominated by any element in D*

*. Since the claim is not always true, the proof is incomplete.*

^{′}In the following, we give an alternative proof that MIXED DOMINA-
TION is NP-complete for splits graphs. The proof involves a transformation
*from modified vertex cover, which may be deﬁned as follows:*

MODIFIED VERTEX COVER

*INSTANCE: A graph G = (V, E) of n vertices and a nonnegative integer k*

*G* *G**’ *
*a*

*b*
*c*
*d*

*a*
*b*
*c*
*d*

*e=ab*
*f=bc*
*g=cd*

Figure 2: A counterexample to the proof in [22]. The collection of shaded elements forms a mixed dominating set in the graph.

*with n + k odd and at least one isolated vertices.*

*OUTPUT: “Yes” if G has a vertex cover of size k, “no” otherwise.*

**Theorem 4.1. MODIFIED VERTEX COVER is NP-complete.**

*Proof. Clearly MODIFIED VERTEX COVER belongs to NP. The correct-*
*ness follows from the fact that C is a vertex cover of G if and only if it is a*
*vertex cover of G∪ K*1, and a trivial reduction from VERTEX COVER.

**Theorem 4.2. ([22]) MIXED DOMINATION is NP-complete for split graphs.**

*Proof. Clearly MIXED DOMINATION belongs to NP. We construct a polynomial-*
time reduction from MODIFIED VERTEX COVER. Given an instance of
*MODIFIED VERTEX COVER, construct a graph G*^{′}*= (V*^{′}*, E*^{′}*) with V** ^{′}* =

*V*

*∪ E and E*

*=*

^{′}*{ uv : u ̸= v, u, v ∈ V } ∪ { ve: v ∈ V, e ∈ E, v ∈ e }. Clearly*

*the graph G*

*can be constructed in linear time in*

^{′}*|V | and |E|. We claim that*

*G has a vertex cover of size k if and only if G*

*has a mixed dominating set of size*

^{′}

^{n+k}_{2}

*.*

^{−1}*Assume that G has a vertex cover C of size k. Since n− k is odd, we can*
*pair the vertices of V* *− C into* ^{n}^{−k−1}_{2} pairs with one vertex left (choose to be
*an isolated vertex of G). These* ^{n}^{−k−1}_{2} *pairs form a set C*^{′}*of edges in G** ^{′}*. It

*is easy to see that C∪ C*

^{′}*form a mixed dominating set of G*

*of size*

^{′}

^{n+k}_{2}

*.*

^{−1}*On the other hand, suppose G*^{′}*has a mixed dominating set D of size*

*n+k**−1*

2 *. For any e = uv* *∈ E ⊆ V*^{′}*with e* *∈ D, either E**G*^{′}*[u]∩ D ̸= ∅ or*
*E*_{G}*′**[v]∩ D ̸= ∅, for otherwise E**G*^{′}*[u]* *∩ D = E**G*^{′}*[v]∩ D = ∅ imply that*
*uv* *∈ E*^{′}*is not mixed dominated by any element in D. In this case, we*

*may assume that E*_{G}*′**[u]∩ D ̸= ∅ and so (D − { e }) ∪ { v } is also a mixed*
*dominating set. Hence we may assume that D∩ E = ∅.*

*Now for any e* *∈ E ⊆ V*^{′}*, either u* *∈ D or ue ∈ D for some u ∈ e. We*
*collect all of these u and ue to form a subset D*^{′}*⊆ D and collect all of these u*
*(includes the u with ue∈ D*^{′}*) to form a subset C* *⊆ V . (If u ∈ D and ue ∈ D*
*for some e∈ E and u ∈ e, then just replace u with v where e = uv.) Notice*
*that C is a vertex cover of G, and* *|C| = |D*^{′}*|. In order to mixed dominate all*
*edges whose end-vertices are in V* *− C, it is the case that |D − D*^{′}*| ≥* ^{n}^{−|C|−1}_{2}
and so ^{n+k}_{2}* ^{−1}* =

*|D| = |D − D*

^{′}*| + |D*

^{′}*| ≥*

^{n}

^{−|C|−1}_{2}+

*|C| which gives k ≥ |C|.*

*Adding enough vertices to C results a vertex cover of G of size k.*

The theorem then follows from Theorem 4.1.

**5. Primal-dual approach for the mixed domination problem in trees**
Although we have presented Algorithm 3 for ﬁnding a minimum mixed
dominating set in a tree, it is still desire to design an algorithm without
using the labeling method, as it needs extra space to store the labels for
each element of a graph. In the following, we present a simple, nonlabeling
algorithm that can compute a minimum mixed dominating set in a tree.

The most beautiful technique used in domination may be the primal-dual
approach. In this technique, besides the original mixed domination problem,
*the following dual problem is also considered. In a graph G = (V, E), a 2-*
*stable set is a subset S* *⊆ V in which every two distinct vertices u and v have*
*distance d(u, v) > 2. The 2-stability number s*_{2}*(G) of G is the maximum*
*size of a 2-stable set in G. Since we are handling both vertices and edges*
concurrently in mixed domination, a parameter which can be viewed as the

*“mixed version” of 2-stable set is introduced. Let ε*_{1} *and ε*_{2} be two elements
*in a graph G = (V, E). Deﬁne the distance between ε*_{1} *and ε*_{2} by

*d*_{m}*(ε*_{1}*, ε*_{2}) =

*d(ε*_{1}*, ε*_{2}*),* *if ε*_{1}*, ε*_{2} *∈ V ;*

min*{ d(x, y): x ∈ ε*1*, y* *∈ ε*2*} , if ε*1 *∈ E or ε*2 *∈ E.*

*A 2-mix-stable set of a graph G = (V, E) is a subset S* *⊆ V ∪ E in which*
*every two distinct elements ε*_{1}*, ε*_{2} *of S satisﬁes the 2-mix-stability:*

*(i) d*_{m}*(ε*_{1}*, ε*_{2}*) > 2, if ε*_{1}*, ε*_{2} *∈ V ;*

*(ii) d*_{m}*(ε*_{1}*, ε*_{2})*≥ 2, if ε*1 *∈ E or ε*2 *∈ E.* (2-mix-stability)

*The 2-mix-stability number ms*_{2}*(G) of G is the maximum size of a 2-mix-*
*stable set in G. Clearly, a 2-stable set in a graph is also a 2-mix-stable set.*

The following lemma shows that in a graph, the cardinality of any 2-mix- stable set is no larger than the cardinality of any mixed dominating set.

**Lemma 5.1. In a graph G = (V, E), if S is a 2-mix-stable set and D is a***mixed dominating set, then* *|S| ≤ |D|.*

*Proof. Deﬁne a function f from S to D by mapping each element ε in S into*
*some mixed neighbor of ε in D. We now claim that f is one-to-one. Suppose*
*two distinct elements ε*_{1} *and ε*_{2} *of S are mapped into the same element b** _{ε}*.

*Case 1 : ε*1

*, ε*2

*∈ V . If b*

*ε*

*∈ V , then clearly d*

*m*

*(ε*1

*, ε*2)

*≤ d(ε*1

*, b*

*ε*

*) + d(ε*2

*, b*

*ε*)

*≤*

*2; if b*

_{ε}*∈ E, then d*

*m*

*(ε*

_{1}

*, ε*

_{2})

*≤ 1. Case 2 : ε*1

*∈ V, ε*2

*∈ E. Then d*

*m*

*(ε*

_{1}

*, ε*

_{2})

*≤*

*d*

_{m}*(ε*

_{1}

*, b*

*)*

_{ε}*≤ 1. Case 3 : ε*1

*, ε*

_{2}

*∈ E. If b*

*ε*

*∈ V , then d*

*m*

*(ε*

_{1}

*, ε*

_{2}

*) = 0; if b*

_{ε}*∈ E,*

*then d*

*m*

*(ε*1

*, ε*2

*) = 1. Since ε*1

*̸= ε*2

*and S is a 2-mix-stable set, we have the*claim and therefore

*|S| ≤ |D| holds.*

**Corollary 5.2. ms**_{2}*(G)≤ γ**m**(G) for any graph G.*

It should be noticed that the inequality of Corollary 5.2 can be strict, as
*shown by the n-cycle C** _{n}* that

ms_{2}*(C** _{n}*) =

⌊_{2n}

5

⌋*, if n̸≡ 0 (mod 5);*

⌈_{2n}

5

⌉*, if n≡ 0 (mod 5),*

*while γ*_{m}*(C** _{n}*) = ⌈

_{2n}5

⌉ *for n≥ 3.*

*For a tree T , an algorithm which outputs a mixed dominating set D and*
*a 2-mix-stable set S with|D| ≤ |S| is designed. From Corollary 5.2, we have*

*|S| ≤ ms*2*(T )≤ γ**m**(G)* *≤ |D| ,*

*which implies that all inequalities are equalities. As a result, D is a minimum*
*mixed dominating set and S is a maximum 2-mix-stable set.*

*The algorithm starts from any penultimate vertex v. At each iteration*
*of the algorithm, penultimate v (resp. vu) is added to D if there exists a*
*pendant vertex (resp. pendant edge) adjacent (resp. incident) to v that is*
*not mixed dominated by any element in D. Besides, the pendant vertex*
*(resp. pendant edge) is also put into S.*

**Algorithm 4. MixDomTreePD**
**Input: A tree T .**

**Output: A minimum mixed dominating set D and a maximum***2-mix-stable set S of T .*

1 *T*^{′}*← T ;*

2 *D← ∅;*

3 *S← ∅;*

4 **while** *|T*^{′}**| > 2 do**

5 *let v be a vertex in T*^{′}*with neighborhoods u, u*_{1}*, . . . , u** _{r}* such that
deg

_{T}*′*

*(u*

*) = 1 for all 1*

_{i}*≤ i ≤ r;*

6 **if** *∃ u**i* *such that N*_{m}*[u** _{i}*]

**∩ D = ∅ then**7 *D← D ∪ { v } and S ← S ∪ { u**i**};*

8 **else**

9 **if** *∃ u**i**v such that N*_{m}*[u*_{i}*v] ∩ D = ∅ then*

10 *D← D ∪ { vu } and S ← S ∪ { u**i**v};*

11 *T*^{′}*← T*^{′}*\ { u*1*, u*_{2}*, . . . , u*_{r}*};*

12 **end**

13 **if** *∃ an element ε in T*^{′}*such that N*_{m}*[ε] ∩ D = ∅ then*

14 *D← D ∪ { ε } and S ← S ∪ { ε };*

**Theorem 5.3. Algorithm MixDomTreePD finds a minimum mixed dominat-***ing set D and a maximum 2-mix-stable set S of a tree T with* *|D| = |S|.*

*Proof. To verify the algorithm, it is suﬃcient to show that D is a mixed*
*dominating set, S is a 2-mix-stable set and* *|D| ≤ |S|. By the construction*
*of the algorithm, D is obviously a mixed dominating set.*

We now show that *|D| = |S| at the end of the algorithm. Suppose*
*elements ε*_{D}*and ε*_{S}*are added to D and S after the i-th iteration of the*
*algorithm, respectively. Then at the beginning of the i-th iteration of the*
*algorithm, ε**S* *is clearly a new element to S. In addition, ε**D* must not be in
*D, for otherwise, ε*_{S}*should not be added to S. Hence, after each iteration*
*of the algorithm, new elements are added to D and S concurrently, and thus*

*|D| = |S| at the end of the algorithm.*

Suppose at the end of the algorithm*|D| = |S| = k. Let ε*1*, ε*_{2}*, . . . , ε** _{k}* and

*s*

_{1}

*, s*

_{2}

*, . . . , s*

_{k}*be the ordered list of elements added to D and S, respectively.*

*Suppose on the contrary that S contains two elements s**i* *and s**j* *with i < j*
*such that s*_{i}*and s*_{j}*violate the 2-mix-stability. Let T** ^{′}* be the tree at the

*beginning of the iteration of adding ε*

_{i}*and s*

_{i}*to D and S, respectively. Then*