5.3 Developing an MDS-silent algorithm
The purpose of this section is to discuss the development of a self-stabilizing MDS-silent algorithm.
The algorithms in both [21] and [76] operate with a central daemon; we now propose an algorithm that operates with a distributed daemon. In particular, we propose a self-stabilizing MDS-silent algorithm Well2n under an unfair distributed daemon in a distance-2 model. Our algorithm uses a two-valued variable state, which has values IN and OUT. We use the same terminology as in Section 3.2 unless otherwise indicated.
To formally define the rules of Well2n, three predicates defined for each node v are needed: noInN br, noBtN br2, and noDpN br2; see Algorithm 7. Notice that noBtN br2 and noDpN br2 require distance-2 knowledge since v has to check the states of its neighbor’s neighbors. Algorithm Well2n uses only two simple rules: rule S1 is regarded as the entering rule and rule S2, the leaving rule.
Algorithm 7 Well2n variables
v.state ∈ {IN, OUT}; // S ={v : v.state = IN}
macros
noInN br(v)≢ ∃w ∈ N(v) : w.state = IN;
noBtN br2(v)≢ ∃w ∈ N(v) : w.state = OUT ∧ w.id < v.id ∧ w has no IN neighbor;
noDpN br2(v)≢ ∃w ∈ N(v) : w.state = OUT ∧ w has exactly one IN neighbor;
rules
S1: if v.state = OUT ∧ noInNbr(v) ∧ noBtNbr2(v)
then v.state := IN; // enter S
S2: if v.state = IN ∧ ¬noInNbr(v) ∧ noDpNbr2(v)
then v.state := OUT; // leave S
Recall that the output of Algorithm 7 is S = {v : v.state = IN}. An MIS algorithm usually applies the following entering rule and leaving rule: “A node having no neighbor in
CHAPTER 5. THE DISTANCE-2 MODEL 5.3. DEVELOPING AN MDS-SILENT ALGORITHM
S joins S and a node having a neighbor in S leaves S.” Algorithm 7 modifies the entering rule and leaving rule to be: “A node enters S if (i) it has no neighbor in S and (ii) its identifier is smaller than any OUT neighbor having no IN neighbor, and a node leaves S if (a) it has a neighbor in S and (b) every neighbor is either in S or has at least two neighbors in S.”
We now prove the correctness of Algorithm 7. First, by using the method for the cor-rectness of Well4n in Lemma 3.3.1, one can prove that in any configuration in which no node is privileged, the set S is a minimal dominating set of G.
Lemma 5.3.1. In any configuration in which no node is privileged, the set S is a minimal dominating set for G.
Proof. Suppose to the contrary that S is not a minimal dominating set for G. Then either (i) S is not a dominating set or (ii) S is a dominating set but not minimal. Consider case (i). Since S is not a dominating set, there exists an OUT node having no IN neighbor. Let u be such a node with the minimum identifier. Since u has no IN neighbor and no better neighbor, rule S1 is enabled, which is a contradiction. Now consider case (ii). Since S is a dominating set but not minimal, there must exist at least one node u∈ S such that S\{u}
is also a dominating set for G. Then|N(u)∩S| ≥ 1 and |N(w)∩S| ≥ 2 for all w in N(u)\S.
Since both ¬noInNbr(u) and noDpNbr2(u) are true, rule S2 is enabled on node u, which is a contradiction.
We now show that Algorithm Well2n is MDS-silent. Note that in each node, rule S1 is not enabled if S is dominating, and rule S2 is not enabled if S is minimal.
Lemma 5.3.2. Algorithm 7 will not make any move if the initial configuration conforms to MDS.
Proof. Recall that Well2n modifies the entering rule and leaving rule of an MIS algorithm to be: “A node enters S if (i) it has no neighbor in S and (ii) its identifier is smaller than
CHAPTER 5. THE DISTANCE-2 MODEL 5.3. DEVELOPING AN MDS-SILENT ALGORITHM
any OUT neighbor having no IN neighbor, and a node leaves S if (a) it has a neighbor in S and (b) every neighbor is either in S or has at least two neighbors in S.” Suppose the initial configuration conforms to MDS. Let S be the set of nodes with state = IN. First consider an arbitrary node u in S. The only rule can be enabled on u is the leaving rule. Since S is an MDS, it is impossible that u has an IN neighbor but has no dependent neighbor; otherwise S\{u} is also a dominating set for G. Thus, at least one of (a) and (b) is false and the leaving rule cannot be enabled on node u. Next consider an arbitrary node w not in S. The only rule that can be enabled on w is the entering rule. Since S is an MDS, it is impossible that w has no IN neighbor. Thus, (i) is f alse and the entering rule cannot be enabled on node w. We have this lemma.
It is not difficult to see that if a node executes the entering rule, then it enters S and will never leave S afterward; furthermore, neighbors of this node will not enter S. See the following lemma.
Lemma 5.3.3. If a node executes rule S1, then it will never leave. Furthermore, neighbors of this node will not enter the set S.
Proof. Let v be a node that executes rule S1. At this moment all neighbors of v have state OUT and with identifier larger then v. Thus, none of these neighbors can execute rule S1 or rule S2. After v enters S, its neighbors have at least one IN neighbor v so they cannot execute rule S1. The only rule that v can execute next is rule S2, but in order to do so, one neighbor of v has to change into state IN first. As long as v is in state IN this is impossible.
Therefore, v will never execute any rule, and its neighbors will not move, either.
We now give the main theorem of this section.
Theorem 5.3.4. Algorithm 7 is self-stabilizing and MDS-silent under an unfair distributed daemon in the distance-2 model. It stabilizes after at most 2n− 1 moves with a minimal dominating set, where n is the number of nodes.
CHAPTER 5. THE DISTANCE-2 MODEL 5.3. DEVELOPING AN MDS-SILENT ALGORITHM
Proof. The theorem follows from Lemmas 5.3.1, 5.3.2, and 5.3.3. The “minus one” part comes from the fact that in a connected graph of order larger than 1, the size of any MDS is at most n− 1.
To see 2n− 1 in Theorem 5.3.4 is tight, consider the star graph of order n with initial state of each node to be IN. Before ending this section, we would like to point out that a preliminary version of this thesis has discussed how to execute our MDS-silent algorithm;
see [11]. In particular, a transformation from the distance-2 model to the distance-1 model is needed. Although the transformers described in [21] and [76] provide ways to transform from an algorithm assuming distance-2 model and central daemon to the distance-1 model, the output algorithm is still not MDS-silent even when we relax the definition of stableness to consider only “membership moves”.
Chapter 6
The signed star domination number of Cayley graphs
The purpose of Chapter 6 is to find out the signed star domination number (a variant of the domination number) of Cayley digraphs and Cayley graphs. A function f : E(G)→ {−1, 1}
is called a signed star dominating function (SSDF) on G if ∑
e∈E(v)f (e) ≥ 1 for every v ∈ V (G), where E(v) is the set of all edges incident with v. The signed star domination number of G is defined as γSS(G) = min{∑
e∈E(G)f (e)| f is an SSDF on G}. In this chapter, we obtain exact values for the signed star domination number of all Cayley digraphs CayD(Γ, S) and certain classes of Cayley graphs Cay(Γ, Ω). Throughout this chapter, we write {u, v} for an edge with endpoints u and v if undirected graphs are considered. For directed graphs, we write (u, v) for “there is an arc from u to v”, that is, the first vertex of the ordered pair is the tail of the arc, and the second is the head.
This chapter is organized as follows. In Section 6.1, we give basic definitions and re-view previous results. In Sections 6.2 and 6.3, we study the signed star domination of the Cayley digraphs and Cayley graphs, respectively. In Section 6.4, we study the signed star domination of {2, 1}-factorable graphs (defined later). A preliminary version of this
chap-55
CHAPTER 6. THE SIGNED STAR DOMINATION 6.1. DEFINITIONS AND PREVIOUS RESULTS
ter appeared as [9]. Note that these solutions are from a joint work with Chelvam and Kalaimurugan.
6.1 Definitions and previous results
Our graph terminologies are standard unless otherwise indicated; see [35, 79]. Let G be a simple connected graph with vertex set V (G) and edge set E(G). For a vertex v ∈ V (G), let E(v) ={{u, v} ∈ E(G) | u ∈ V (G)}. A function f : E(G) → {−1, 1} is called a signed star dominating function (SSDF) on G if ∑
e∈E(v)f (e)≥ 1 for every v ∈ V (G). The signed star domination number of G is defined as γSS(G) = min {∑
e∈E(G)f (e) | f is an SSDF on G} and such an f attaining the minimum value is called a minimum SSDF on G. Note that γSS(G) is well-defined only if G contains no isolated vertex. A set {f1, f2, . . . , fd} of SSDFs on G with the property that ∑d
i=1fi(e) ≤ 1 for each e ∈ E(G) is called a signed star dominating family (of functions) on G. The maximum number of functions in a signed star dominating family on G is called the signed star domatic number of G and is denoted by dSS(G). Let D be a digraph with vertex set V (D) and arc set A(D). For each vertex v ∈ V (D), let A(v) be the set of all out-going arcs from v. By replacing E(v) by A(v), one can define the SSDF on D and γSS(D) = min{∑
a∈A(D)f (a)| f is an SSDF on D}.
Let Γ be a finite nontrivial group with the identity element ι and S be a nonempty subset of Γ. The Cayley digraph CayD(Γ, S) is the digraph whose vertices are the elements of Γ, and there is an arc from α to ασ whenever α∈ Γ and σ ∈ S. A subset S of Γ is symmetric if σ−1 ∈ S whenever σ ∈ S. A generating set of Γ is a subset that is not contained in any proper subgroup of Γ. Let Ω be a symmetric generating subset of nonidentity elements of Γ. The Cayley graph Cay(Γ, Ω) corresponding to Γ and Ω is the ordinary graph with vertex set Γ and edge set E ={{α, ασ} | α ∈ Γ, σ ∈ Ω}. Note that in the Cayley graph, the edges α to ασ and ασ to α are considered one and the same.