Algorithmic aspect of stratified domination in graphs

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

www.math.ntu.edu.tw/ ~ mathlib/preprint/2012- 02.pdf

Algorithmic aspect of stratified domination in graphs

Gerard Jennhwa Chang, Chan-Wei Changd, David Kuod, and Sheung- Hung Poon

March 21, 2012

Algorithmic aspect of

stratified domination in graphs

Gerard Jennhwa Chang

Chan-Wei Chang

David Kuo

Sheung-Hung Poon

aDepartment of Mathematics, National Taiwan University, Taipei 10617, Taiwan

bTaida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan

cNational Center for Theoretical Sciences, Taipei Office, Taiwan

dDepartment of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan

eDepartment of Computer Science, National Tsing Hua University, Hsinchu, Taiwan

March 21, 2012

Abstract

Chartrand, Haynes, Henning and Zhang introduced a variation of domina- tion called stratified domination in graphs. This paper studies stratified domi- nation from an algorithmic point of view. A 2-stratified (or black-white) graph is a graph in which every vertex is colored black or white. Given a black-white graph F rooted at a white vertex v, an F-coloring coloring of a graph G is a black-white coloring of V(G) for which every white vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. An F-dominating set of G is the set of all black vertices in an F-coloring. The F-domination number γF(G) of G is the minimum cardinality of an F-dominating set. We consider the 3-vertex black-white graph F3rooted at a white vertex v adjacent to another white ver- tex u, which adjacent to a black vertex w. We prove that the F3-domination problem is NP-complete for bipartite graphs, for planar graphs and for chordal graphs. We also give a linear-time algorithm for the F3-domination problem in trees.

Keywords: k-stratified graph, domination, F-domination, NP-complete, bipar- tite graph, planar graph, chordal graph, tree.

∗E-mail: gjchang@math.ntu.edu.tw. Supported in part by the National Science Council under grant NSC98-2115-M-002-013-MY3.

†E-mail:d9411001@ems.ndhu.edu.tw.

‡E-mail:davidk@mail.am.ndhu.edu.tw. Supported in part by the National Science Council under grant NSC97-2115-M-259-002-MY3.

§E-mail: spoon@cs.nthu.edu.tw. Supported in part by the National Science Council under grants NSC97-2221-E-007-054-MY3 and NSC99-2218-E-007-016.

1 Introduction

The study of combining stratification and domination in graphs was started by Char- trand, Haynes, Henning and Zhang [5]. A k-stratified graph is a graph G together with a partition of its vertex set V(G) into k nonempty subsets, called strata or color classes. In this paper, we use black and white for colors in 2-stratified graphs.

Rashidi [22] studied a number of problems involving stratified graphs; while dis- tance in stratified graphs was investigated in [2, 3, 6].

In a graph G, a vertex v dominates itself and all neighbors which are vertices adjacent to v. A dominating set of G is a subset D⊆ V(G) for which every vertex of V(G)\ D is dominated by some neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A γ-set of G is a dominating set of cardinality γ(G). Domination in graphs and its variations have been well studied in the literature, see the books by Haynes, Hedetniemi and Slater [15, 16].

Among the variations of domination, we mention three. A total dominating set of G is a subset D ⊆ V(G) for which every vertex of V(G) is dominated by some neighbor in D. A restrained dominating set of G is a subset D ⊆ V(G) for which every vertex of V(G)\D is dominated by some neighbor in D and by some neighbor not in D. A k-dominating set is a subset D⊆ V(G) for which every vertex of V(G)\D is dominated by at least k neighbors in D. The total domination number γ_t(G), the restrained domination number γ_r(G) and the k-domination numbers γk(G) are the minimum cardinality of a total dominating set, a restrained dominating set and a k-dominating set, respectively. Total domination in graphs was introduced by Cock- ayne, Dawes and Hedetniemi [7], restrained domination by Telle and Proskurowski [23] and k-domination by Fink and Jacobson [8].

In [5] a new framework for studying domination was presented. Suppose F is a 2-stratified graph with a specified white vertex v, called the root of F. An F-coloring of a graph G is a black-white coloring of V(G) for which every white vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. An F-dominating set of G is the set of all black vertices in an F-coloring. The F-domination number γF(G) of G is the minimum cardinality of an F-dominating set. If G has no copy of F, then γF(G) = |V(G)|. A γF-set is a γ_F-dominating set of cardinality γ_F(G). See [4, 10, 11, 12, 13, 14, 18, 19, 20, 21] for further study and [17] for a survey on F-domination.

An equivalent definition for F-domination is as follows. A subset D ⊆ V(G) corresponds to a black-white coloring f_D of G for which D is the set of all black vertices. For the subset D, a vertex v of G is F-dominated itself and a black vertex in a copy of F rooted at v in G using the black-white coloring f_D. Using this new term, an F-dominating set of G is then a subset D⊆ V(G) for which every vertex of

V(G)\ D is F-dominated by some vertex of D.

If F0 is a K2 rooted at a white vertex v adjacent to a black vertex, then F0- domination is the same as the usual domination. Hence, γ_F0(G) = γ(G). For the case when F is a P3, there are five 2-stratified graphs, see Figure 1.

ve

u

u

F1

ve

u

e

F2

ve

e

u

F3

ve







DD DD u DDe

F4

ve







DD DD u DDu

F5

Figure 1: The five 2-stratified graphs P3.

It was proved in [5] that for any connected graph G of at least three vertices, γF₁ = γt(G), γF₂ = γ(G), γF₄ = γr(G) and γF₅ = γ2(G). The parameter γF₃

appears to be new. It then was studied in the literature. All the results so far are on upper and lower bounds and exact values for special graphs. The attempt of this paper is to study F3-domination from an algorithmic point of view. In particular, we prove that the F₃-domination problem is NP-complete for bipartite graphs, for planar graphs and for chordal graphs. We also give a linear-time algorithm for the F3-domination problem in trees.

We remark that a subset D ⊆ V(G) is an F3-dominating set of G if every vertex v∈ V(G) \ D is dominated by some vertex u ∈ V(G) \ D and u in turn is dominated by some vertex w∈ D.

2 NP-completeness for the F

-domination problem

This section proves that the F3-domination problem is NP-complete on bipartite graphs, planar graphs and chordal graphs.

Theorem 1 The F3-domination problem is NP-complete on bipartite graphs and pla- nar graphs.

Proof. It is clear that the F₃-domination problem is in NP. We shall show that the F3-domination problem is NP-hard on bipartite graphs (respectively, planar graphs) by reducing the vertex cover problem to it. For a given graph G, the vertex cover problem asks for a vertex cover, which is a subset C⊆ V(G) for which every edge of G has at least one end vertex in C, of size at most k in G. We may assume that

G has no isolated vertex. Let V(G) = {v1, v2, . . . , v_n}. In the reduction, a bipartite graph G^′ is constructed from G as follows. (In the case of G is planar, G^′ is also planar.) For each edge e ={vi, v_j} of G with i < j, we create a representative vertex u^e and replace the edge e by a 2-path (vi, u^e, vj). We further connect u^e to a new degree-1 vertex x^e and a new 5-path (p^e, q^e, r^e, s^e, t^e), see Figure 2.

e

vie ee vj

e

G

e

b b b b b b b

vie

x_b^e _bu^e p_b^e q_b^e r_b^e s_b^e t_b^e vj e

b b b b b b b

e

G^′

Figure 2: The graph G^′ constructed from G.

We claim that G has a vertex cover of size at most k if and only if G^′ has an F3-dominating set of size at most k + 2|E(G)|.

Suppose that G has a vertex cover C of size at most k. Consider the vertex set D = C∪ {q^e, t^e: e ∈ E(G)} in G^′. First, |D| = |C| + 2|E(G)| 6 k + 2|E(G)|. Next, for each e ∈ E(G), we have that r^e, s^e and u^e are F3-dominated by t^e, q^e and q^e, respectively. Also for each e ∈ E(G), since C is a vertex cover, e has an end vertex in C which F3-dominates x^e and p^e. Finally, each vertex x ∈ V(G) is in an edge e and so one of its end vertices is in C and F3-dominates x. These show that D is an F3-dominating set of G^′of size at most k + 2|E(G)|.

Conversely, suppose that G^′ has an F3-dominating set D of size at most k + 2|E(G)|. Let C = (D ∩ V(G)) ∪ {vi: e = {vi, vj}, i < j, D ∩ {x^e, p^e} ̸= ∅}. For each e∈ E(G), since D contains at least two vertices in {q^e, r^e, s^e, t^e}, it is the case that

|C| 6 k. Next, consider any edge e = {vi, vj} of G. Since x^e is F-dominated by x^e, p^e, vi or vj, at least one of vi, vj is in C. These prove that C is a vertex cover of G of size at most k.

The NP-completeness of the F3-domination problem on bipartite (respectively, planar) graphs then follows from the NP-completeness of the vertex cover problem

on general (respectively, planar) graphs. 

Theorem 2 The F3-domination problem is NP-complete on chordal graphs.

Proof. It is clear that the F3-domination problem is in NP. We shall show that the F3-domination problem is NP-hard on chordal graphs by reducing the domination problem on chordal graphs to it. For a given chordal graph G, the domination

problem asks for a dominating set of size at most k in G. In the reduction, a chordal graph G^′ is constructed from G as follows. For each vertex v of G, we create two new adjacent vertices v^′and v^′′both adjacent to v. For each edge e ={u, v} of G, we create two new adjacent vertices u^e and v^e both adjacent to u and v, see Figure 3.

e

ue ee v

e

G

e b```b b

HHHb

@u@e b```b

u^eb_HHH^b_@v^e

@e

e bv^′

```bv<sup>′′</sup>

bv

HHHb

@@e b```b

G<sup>′</sup>

Figure 3: The graph G<sup>′</sup> constructed from G.

We claim that G has a dominating set of size at most k if and only if G<sup>′</sup> has an F3-dominating set of size at most k.

Suppose that G has a dominating set D of size at most k. We shall check that D is an F3-dominating set of G<sup>′</sup>. We first consider v<sup>′</sup> and v<sup>′′</sup> for v ∈ V(G). If v ∈ D, then v<sup>′</sup>and v<sup>′′</sup>are F3-dominated by v. If v /∈ D, then it is dominated by some u ∈ D, which F3-dominates v<sup>′</sup> and v<sup>′′</sup>. We next consider u<sup>e</sup> and v<sup>e</sup> for e = {u, v} ∈ E(G).

Since v∈ D or v /∈ D but is dominated by a neighbor w, one of them F3-dominates u<sup>e</sup> and v<sup>e</sup>. Finally, each v ∈ V(G) is dominated by v or a neighbor w, so one of them F3-dominates v. These show that D is an F3-dominating set of G<sup>′</sup>.

Conversely, suppose that G<sup>′</sup> has an F<sub>3</sub>-dominating set D<sup>′</sup> of size at most k. Let D ={v: {v, v<sup>′</sup>, v<sup>′′</sup>, v<sup>e</sup>: v∈ e} ∩ D<sup>′</sup> ̸= ∅}. Then |D| 6 |D<sup>′</sup>| 6 k. We shall check that D is a dominating set of G. For any vertex v ∈ V(G) \ D, we have that v, v<sup>′</sup>, v<sup>′′</sup>, v<sup>e</sup> ∈ D/ <sup>′</sup> and so v<sup>′</sup> is F3-dominated by a neighbor u or u<sup>e</sup> of v. Then u∈ D and dominates v.

Hence, D is a dominating set of G of size at most k.

The NP-completeness of the F<sub>3</sub>-domination problem on chordal graphs then fol- lows from the NP-completeness of the domination problem on chordal graphs.  While the domination problem is NP-complete on split graphs, the F3-domination problem is polynomially solvable for split graphs, which are graphs whose vertex set can be partitioned into a stable set and a clique.

3 Algorithm for trees

This section establishes a linear-time algorithm for the F3-domination problem on trees by using a dynamic algorithm.

In a graph G, a subset D⊆ V(G) F3-dominates another subset S⊆ V(G) if every vertex v∈ S \ D is F-dominated by some vertex u ∈ D.

We consider the graph G rooted at a specified vertex v and denote it by G<sub>v</sub>. The composition of two disjoint rooted graphs G<sub>v</sub> and H<sub>u</sub> is the graph G<sub>v</sub>⊕ Hu rooted at v obtained from the union of G<sub>v</sub> and H<sub>u</sub> by adding an edge{v, u}, see Figure 4.

Iv

u u

u u

u<sub>u</sub>



 BB

BBB

Hu

v<sub>u</sub>



 BB

BBB PPPPP

Gv

Figure 4: I<sub>v</sub> = Gv⊕ Hu.

A tee can be obtained from isolated vertices by a sequences of the⊕ operations.

Hence, to establish an algorithm for trees, we only need to establish formula for γF<sub>3</sub>(Gv ⊕ Hu) in terms of γF<sub>3</sub>(Gv) and γF<sub>3</sub>(Hu). To do so, we in fact solve more variant problems with boundary conditions. More precisely, we consider γ<sub>F3</sub>(Gv, ij) for ij ∈ {1, 0, 0, 11, 10, 01, 01, 00, 00, 00, 00}, where i ∈ {1, 0, 0} and j ∈ {1, 0, 0, NIL}. An ij-F3-dominating set of a rooted graph G<sub>v</sub> is a subset D ⊆ V(G) satisfying the following three conditions. Recall that N(v) is the set of all neighbors of v and N[v] ={v} ∪ N(v). We write NG(v) and NG[v] when the graph G is relevant.

C1. v∈ D when i ∈ {1, 1} and v /∈ D when i ∈ {0, 0}.

C2. N(v)∩ D ̸= ∅ when j ∈ {1, 1} and N(v) ∩ D = ∅ when j ∈ {0, 0}.

C3. D F3-dominates S, where S = V(G)\ N[v] for ij = 00, S = V(G) \ N(v) for i ̸= 0 but j = 0, S = V(G) \ {v} for i = 0 but j ̸= 0, and S = V(G) otherwise.

Notice that we don’t need to consider 1j, 01 or 01 as it in fact is the same as 1j, 01 or 01, respectively. We also don’t need 11 and 10 in the recursive formula. The ij-F3-domination number γ<sub>F3</sub>(G, ij) of a rooted graph Gv is the minimum cardinality of an ij-F3-dominating set.

Theorem 3 If Gv and H<sub>u</sub> are two disjoint rooted trees whose composition is I<sub>v</sub>, then the following holds.

(1) γ<sub>F3</sub>(Iv) =min{γF<sub>3</sub>(Iv, 1), γ<sub>F3</sub>(Iv, 0)}.

(2) γ<sub>F3</sub>(Iv, i) = min{γF<sub>3</sub>(Iv, i1), γ<sub>F3</sub>(Iv, i0)} for i ∈ {1, 0, 0}.

(3) γ<sub>F3</sub>(Iv, 11) = min{γF<sub>3</sub>(Gv, 11) + γ<sub>F3</sub>(Hu, 1), γ<sub>F3</sub>(Gv, 11) + γ<sub>F3</sub>(Hu, 01), γF<sub>3</sub>(Gv, 11) + γ<sub>F3</sub>(Hu, 00), γ<sub>F3</sub>(Gv, 10) + γ<sub>F3</sub>(Hu, 1)}.

(4) γ<sub>F3</sub>(Iv, 10) = min{γF<sub>3</sub>(Gv, 10) + γ<sub>F3</sub>(Hu, 01), γ<sub>F3</sub>(Gv, 10) + γ<sub>F3</sub>(Hu, 00)}.

(5) γ<sub>F3</sub>(Iv, 01) = min{γF<sub>3</sub>(Gv, 01) + γ<sub>F3</sub>(Hu, 1), γ<sub>F3</sub>(Gv, 00) + γ<sub>F3</sub>(Hu, 1), γF<sub>3</sub>(Gv, 01) + γ<sub>F3</sub>(Hu, 01), γ<sub>F3</sub>(Gv, 01) + γ<sub>F3</sub>(Hu, 00)}.

(6) γ<sub>F3</sub>(Iv, 01) = min{γF<sub>3</sub>(Gv, 01) + γ<sub>F3</sub>(Hu, 1), γ<sub>F3</sub>(Gv, 00) + γ<sub>F3</sub>(Hu, 1), γF<sub>3</sub>(Gv, 01) + γ<sub>F3</sub>(Hu, 0)}.

(7) γ<sub>F3</sub>(Iv, 0j) = min{γF<sub>3</sub>(Gv, 0j) + γ<sub>F3</sub>(Hu, j1), γ<sub>F3</sub>(Gv, 0j) + γ<sub>F3</sub>(Hu, j0)} for j ∈ {0, 0}.

(8) γ<sub>F3</sub>(Iv, 0j) = γ<sub>F3</sub>(Gv, 0j) + γ<sub>F3</sub>(Hu, j) for j∈ {0, 0}.

Proof. (1) and (2) are obvious. In the proofs of the following cases, we write a subset D⊆ V(Iv)as the union D<sup>′</sup>∪ D<sup>′′</sup>, where D<sup>′</sup> ⊆ V(Gv)and D<sup>′′</sup>⊆ V(Hu).

(3) For v∈ D and NIv(v)∩D ̸= ∅, there are four cases: v ∈ D<sup>′</sup>and N<sub>Gv</sub>(v)∩D<sup>′</sup> ̸=

∅ with u ∈ D<sup>′′</sup>; v ∈ D<sup>′</sup> and NGv(v)∩ D<sup>′</sup> ̸= ∅ with u /∈ D<sup>′′</sup> and NHu(u)∩ D<sup>′′</sup> ̸= ∅;

v ∈ D<sup>′</sup> and NGv(v)∩ D<sup>′</sup> ̸= ∅ with NHu[u]∩ D<sup>′′</sup> = ∅; v ∈ D<sup>′</sup> and NGv(v)∩ D<sup>′</sup> = ∅ with u∈ D<sup>′′</sup>. The formula then follows from that D is a 11-F3-dominating set of Iv

if and only if D<sup>′</sup> is a 11-F3-dominating set of Gv with D<sup>′′</sup> a 1-F3-dominating set of Hu, or D<sup>′</sup>is a 11-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup>a 01-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 11-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 00-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 10-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 1-F3-dominating set of H<sub>u</sub>.

(4) For v∈ D and NIv(v)∩D = ∅, there are two cases: v ∈ D<sup>′</sup>and NGv(v)∩D<sup>′</sup> =

∅ with u /∈ D<sup>′′</sup> and NHu(u)∩ D<sup>′′</sup> ̸= ∅; v ∈ D<sup>′</sup> and NGv(v)∩ D<sup>′</sup>=∅ with u /∈ D<sup>′</sup> and NHu[u]∩ D<sup>′′</sup> =∅. The formula then follows from that D is a 10-F3-dominating set of I<sub>v</sub> if and only if D<sup>′</sup> is a 10-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 01-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 10-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 00-F3-dominating set of Hu.

(5) For v /∈ D and NIv(v)∩D ̸= ∅, there are four cases: v /∈ D<sup>′</sup>and NGv(v)∩D<sup>′</sup> ̸=

∅ with u ∈ D<sup>′′</sup>; N<sub>Gv</sub>[v]∩ D<sup>′</sup> = ∅ with u ∈ D<sup>′′</sup>; v /∈ D<sup>′</sup> and N<sub>Gv</sub>(v)∩ D<sup>′</sup> ̸= ∅ with u /∈ D<sup>′′</sup>and N<sub>Hu</sub>(u)∩ D<sup>′′</sup> ̸= ∅; v /∈ D<sup>′</sup> and N<sub>Gv</sub>(v)∩ D<sup>′</sup≯= ∅ with NHu[u]∩ D<sup>′′</sup> =∅.

The formula then follows from that D is a 01-F3-dominating set of I<sub>v</sub> if and only if D<sup>′</sup> is a 01-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 1-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 00-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 1-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a

01-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 01-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 01-F3- dominating set of G<sub>v</sub> with D<sup>′′</sup> a 00-F3-dominating set of H<sub>u</sub>.

(6) For v /∈ D and NIv(v)∩D ̸= ∅, there are three cases: v /∈ D<sup>′</sup>and N<sub>Gv</sub>(v)∩D<sup>′</sup> ̸=

∅ with u ∈ D<sup>′′</sup>; N<sub>Gv</sub>[v]∩ D<sup>′</sup> = ∅ with u ∈ D<sup>′′</sup>; v /∈ D<sup>′</sup> and N<sub>Gv</sub>(v)∩ D<sup>′</sup> ̸= ∅ with u /∈ D<sup>′′</sup>. The formula then follows from that D is a 01-F3-dominating set of I<sub>v</sub> if and only if D<sup>′</sup> is a 01-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 1-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 00-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 1-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 01-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a 0-F3-dominating set of H<sub>u</sub>.

(7) For N<sub>Iv</sub>[v] ∩ D = ∅, there are two cases: NGv[v]∩ D<sup>′</sup> = ∅ with u /∈ D<sup>′′</sup>

and N<sub>Hu</sub>(u)∩ D<sup>′′</sup> ̸= ∅; NGv[v]∩ D<sup>′</sup> = ∅ with NHu[u]∩ D<sup>′′</sup> = ∅. The formula then follows from that D is a 0j-F3-dominating set of I<sub>v</sub> for j ∈ {0, 0} if and only if D<sup>′</sup> is a 0j-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a j1-F3-dominating set of H<sub>u</sub>, or D<sup>′</sup> is a 0j-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a j0-F3-dominating set of H<sub>u</sub>.

(8) The formula follows from that D is a 0j-F3-dominating set of I<sub>v</sub> for j∈ {0, 0}

if and only if D<sup>′</sup> is a 0j-F3-dominating set of G<sub>v</sub> with D<sup>′′</sup> a j-F3-dominating set of

Hu. 

The above theorem together with the following initial conditions gives a linear- time algorithm for the F3-domination in trees:

γ(K1, 1) = 1, γ(K1, 0) = γ(K1, 00) = γ(K1, 00) = 0 and γ(K1, ij) =∞ for other ij.

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