Vizing's conjecture: a survey and recent results

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.

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 $A^n$.

Journal of Combinatorial Optimization, 2008

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63-69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ t (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γ t (G)Γ t (H ) ≤ 2Γ t (G H ).

Discrete Applied Mathematics

We show that for any claw-free graph G and any graph H, γ(G H) ≥ 2 3 γ(G)γ(H), where γ(G) is the domination number of G.

Journal of Combinatorial Optimization, 2009

Let γ t {k}(G) denote the total {k}-domination number of graph G, and let $G\mathbin{\square}H$ denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, $\gamma _{t}^{\{k\}}(G)\gamma _{t}^{\{k\}}(H)\le k(k+1)\gamma _{t}^{\{k\}}(G\mathbin{\square}H)$ . As a corollary of this result, we have $\gamma _{t}(G)\gamma _{t}(H)\le 2\gamma _{t}(G\mathbin{\square}H)$ for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005).

2000

In this paper, we study the domination number, the global dom ination number, the cographic domination number, the global co graphic domination number and the independent domination number of all the graph products which are non-complete ...

Electronic Notes in Discrete Mathematics, 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. We investigate a similar problem for paired-domination, and obtain a lower bound in terms of product of domination number of one factor and 3-packing of the other factor. Some results are obtained by applying a new graph invariant called rainbow domination.

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 .

Discussiones Mathematicae Graph Theory, 2012

The total restrained domination number of G, denoted by γ t r (G), is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.

Discrete Applied Mathematics

A set of vertices S in a simple isolate-free graph G is a semi-total dominating set of G if it is a dominating set of G and every vertex of S is within distance 2 or less with another vertex of S. The semi-total domination number of G, denoted by γ t2 (G), is the minimum cardinality of a semi-total dominating set of G. In this paper we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs G and H, γ t2 (G ✷ H) ≥ 1 3 γ t2 (G)γ t2 (H).