Vizing’s conjecture for chordal graphs

Theory and Applications of Graphs, 2016

For any graph G = (V, E), a subset S ⊆ V dominates G if all vertices are contained in the closed neighborhood of S, that is N [S] = V. The minimum cardinality over all such S is called the domination number, written γ(G). In 1963, V.G. Vizing conjectured that γ(G H) ≥ γ(G)γ(H) where stands for the Cartesian product of graphs. In this note, we define classes of graphs An, for n ≥ 0, so that every graph belongs to some such class, and A0 corresponds to class A of Bartsalkin and German. We prove that for any graph G in class A1, γ(G H) ≥ γ(G) − γ(G) γ(H).

For any graph $G=(V,E)$, a subset $S\subseteq V$ $dominates$ $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\gamma(G)$. In 1963, V.G. Vizing conjectured that $\gamma(G \square H) \geq \gamma(G)\gamma(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we prove that if $\left|G\right|\geq \gamma(G)\gamma(H)$ and $\left|H\right|\geq \gamma(G)\gamma(H)$, then the conjecture holds. This result quickly implies Vizing's conjecture for almost all pairs of graphs $G,H$ with $\left|G\right|\geq \left|H\right|$, satisfying $\left|G\right|\leq q^{\frac{\left|H\right|}{\log_q\left|H\right|}}$ for $q=\frac{1}{1-p}$ and $p$ the edge probability of the Erd\H{o}s-R\'enyi random graph.

The Electronic Journal of Combinatorics, 2013

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

Electronic Notes in Discrete Mathematics, 2013

In this work we obtain a new graph class where the {k}-dominating function problem ({k}-DOM) is NP-complete: the class of chordal graphs. We also identify some maximal subclasses for which it is polynomial time solvable. Firstly, by relating this problem with the k-dominating function problem (k-DOM), we prove that {k}-DOM is polynomial time solvable for strongly chordal graphs. Besides, by expressing the property involved in k-DOM in Counting Monadic Second-order Logic, we obtain that both problems are linear time solvable for bounded treewidth graphs. Finally, we show that {k}-DOM is linear time solvable for spider graphs.

Graphs and Combinatorics, 2005

The most famous open problem involving domination in graphs is Vizing's conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.

Journal of Ultra Scientist of Physical Sciences Section A

In11, Kulli and Janakiram initiate the concept of maximal domination in graphs. In this paper, we obtained some bounds and characterizations. Also, we estimate the value of the maximal domination number of some graph products such as join of graphs, corona product, cartesian product and strong product.

Results in Mathematics, 1990

A set S of vertices of a graph G is dominating if each vertex z not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number,-y(G),

2008

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of graph G, if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. It is well known that if e ∈ E(G), then γ(G−e)−1 ≤ γ(G) ≤ γ(G−e). In this paper, as an application of this inequality, we obtain the domination number of some certain graphs.

Journal of Graph Theory, 2012

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper.

Discussiones Mathematicae Graph Theory, 2021

We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to Vizing's conjecture. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of Clark and Suen as follows: γ(X2Y) ≥ max 1 2 γ(X)γ t (Y), 1 2 γ t (X)γ(Y) , where γ stands for the domination number, γ t is the total domination number, and X 2 Y is the Cartesian product of graphs X and Y .