The Number of 2-dominating Sets, and 2-domination Polynomial of a Graph

Abstract: Let $$G=(V,E)$$ be a simple graph. A set $$D\subseteq V$$ is a $$2$$-dominating set of $$G$$, if every vertex of $$V\setminus D$$ has at least two neighbors in $$D$$. The $$2$$-domination number of a graph $$G$$, is denoted by $$\gamma_{2}(G)$$ and is the minimum size of the $$2$$-dominating sets of $$G$$. In this paper, we count the number of $$2$$-dominating sets of $$G$$. To do this, we consider a polynomial which is the generating function for the number of $$2$$-dominating sets of $$G$$ and call it $$2$$-domination polynomial. We study some properties of this polynomial. Furthermore, we compute the $$2$$-domination polynomial for some of the graph families

Dominating sets and domination polynomials of certain graphs, II

The domination polynomial of a graph G of order n is the polynomial [formula] where d(G, i) is the number of dominating sets of G of size i, and ϒ (G) is the domination number of G. In this paper, we obtain some properties of the coefficients of D(G, x). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by G'(m), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs G' (m), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if n ≡ 0, 2(mod 3) and D(G, x) = D(Cn, x), then G = Cn

Dominating Sets and Domination Polynomials of Paths

Let G=(V,E) be a simple graph. A set S⊆V is a dominating set of G, if every vertex in V\S is adjacent to at least one vertex in S. Let 𝒫ni be the family of all dominating sets of a path Pn with cardinality i, and let d(Pn,j)=|𝒫nj|. In this paper, we construct 𝒫ni, and obtain a recursive formula for d(Pn,i). Using this recursive formula, we consider the polynomial D(Pn,x)=∑i=⌈n/3⌉nd(Pn,i)xi, which we call domination polynomial of paths and obtain some properties of this polynomial

Clustering and Domination in Perfect Graphs

A K-Cluster in a graph is an induced sub graph on k-vertices, which maximizes the number of edges. Both the K-Cluster problem and the K-dominating set problem are NP-complete for graphs in general. In this paper we investigate the complexity status of these problems on various subclasses of perfect graphs. In particular, we examine comparability graphs, chordal graphs, bipartite graphs, split graphs, co graphs and K-trees. For example, it is shown that the K-Cluster problem is NP-complete for both bipartite and chordal graphs and the independent K-dominating set problem is NP-complete for bipartite graphs. Furthermore, where the Kcluster problem is polynomial we study the weighted and connected versions as well. Similarly we also look at the minimum K-dominating set problem on families, which have polynomial K-dominating set algorithms.Technical report DCS-TR-13

A polynomial-time algorithm for the paired-domination problem on permutation graphs

AbstractA set S of vertices in a graph H=(V,E) with no isolated vertices is a paired-dominating set of H if every vertex of H is adjacent to at least one vertex in S and if the subgraph induced by S contains a perfect matching. Let G be a permutation graph and π be its corresponding permutation. In this paper we present an O(mn) time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph G with n vertices and m edges

Maker-Breaker domination game

International audienceWe introduce the Maker-Breaker domination game, a two player game on a graph. At his turn, the rst player, Dominator, select a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker-Breaker games, that is studied here in a combinatorial context. In this paper, we rst prove that deciding the winner of the Maker-Breaker domination game is PSPACE-complete, even for bipartite graphs and split graphs. It is then showed that the problem is polynomial for cographs and trees. In particular, we dene a strategy for Dominator that is derived from a variation of the dominating set problem, called the pairing dominating set problem

Improved bottleneck domination algorithms

AbstractW.C.K. Yen introduced BOTTLENECK DOMINATION and BOTTLENECK INDEPENDENT DOMINATION. He presented an O(nlogn+m)-time algorithm to compute a minimum bottleneck dominating set. He also obtained that the BOTTLENECK INDEPENDENT DOMINATING SET problem is NP-complete, even when restricted to planar graphs.We present simple linear time algorithms for the BOTTLENECK DOMINATING SET and the BOTTLENECK TOTAL DOMINATING SET problem. Furthermore, we give polynomial time algorithms (most of them with linear time-complexities) for the BOTTLENECK INDEPENDENT DOMINATING SET problem on the following graph classes: AT-free graphs, chordal graphs, split graphs, permutation graphs, graphs of bounded treewidth, and graphs of clique-width at most k with a given k-expression

Connected Domination in Graphs

A connected dominating set D is a set of vertices of a graph G = (V, E) such that every vertex in V − D is adjacent to at least one vertex in D and the subgraph hDi induced by the set D is connected. The connected domination number γc(G) is the minimum of the cardinalities of the connected dominating sets of G. The problem of finding a minimum connected dominating set D is known to be NP-hard. Many polynomial time algorithms that achieve some approximation factors have been provided earlier in finding a minimum connected dominating set. In this work, we present a survey on known properties of graph domination as well as some approximation algorithms. We implemented some of these algorithms and tested them with random graphs and compared their performance in finding a minimum connected dominating set D. We present the breadth first search algorithm as a heuristic for finding a connected dominating set whose cardinality is hopefully close to that of a minimum connected dominating set. The algorithm finds a spanning tree T of the graph G = (V, E) using breadth first search, and picks up the non-leaf nodes as the connected dominating set D. There are graphs for which the Breadth first search heuristic does not work so well. We implemented some local optimization procedures that would improve the performance of the breadth first search heuristic in finding the minimum connected dominating set D

Connected Domination in Graphs

A connected dominating set D is a set of vertices of a graph G = (V, E) such that every vertex in V − D is adjacent to at least one vertex in D and the subgraph hDi induced by the set D is connected. The connected domination number γc(G) is the minimum of the cardinalities of the connected dominating sets of G. The problem of finding a minimum connected dominating set D is known to be NP-hard. Many polynomial time algorithms that achieve some approximation factors have been provided earlier in finding a minimum connected dominating set. In this work, we present a survey on known properties of graph domination as well as some approximation algorithms. We implemented some of these algorithms and tested them with random graphs and compared their performance in finding a minimum connected dominating set D. We present the breadth first search algorithm as a heuristic for finding a connected dominating set whose cardinality is hopefully close to that of a minimum connected dominating set. The algorithm finds a spanning tree T of the graph G = (V, E) using breadth first search, and picks up the non-leaf nodes as the connected dominating set D. There are graphs for which the Breadth first search heuristic does not work so well. We implemented some local optimization procedures that would improve the performance of the breadth first search heuristic in finding the minimum connected dominating set D