Upper paired domination versus upper domination

A paired dominating set $P$ is a dominating set with the additional property that $P$ has a perfect matching. While the maximum cardainality of a minimal dominating set in a graph $G$ is called the upper domination number of $G$, denoted by $\Gamma(G)$, the maximum cardinality of a minimal paired dominating set in $G$ is called the upper paired domination number of $G$, denoted by $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know that $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any graph $G$ without isolated vertices. We focus on the graphs satisfying the equality $\Gamma_{pr}(G)= 2\Gamma(G)$. We give characterizations for two special graph classes: bipartite and unicyclic graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ by using the results of Ulatowski (2015). Besides, we study the graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and a restricted girth. In this context, we provide two characterizations: one for graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ and girth at least 6 and the other for $C_3$-free cactus graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$. We also pose the characterization of the general case of $C_3$-free graphs with $\Gamma_{pr}(G)= 2\Gamma(G)$ as an open question.

Almost well-dominated bipartite graphs with minimum degree at least two

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

Graphs with equal domination and certified domination numbers

A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number (upper domination number, respectively) of \(G\), denoted by \(\gamma(G)\) (\(\Gamma(G)\), respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\) and every vertex in \(D\) has either zero or at least two neighbors in \(V_G-D\). The cardinality of a smallest (largest minimal, respectively) certified dominating set of \(G\) is called the certified (upper certified, respectively) domination number of \(G\) and is denoted by \(\gamma_{\rm cer}(G)\) (\(\Gamma_{\rm cer}(G)\), respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.

Avoidance colourings for small nonclassical Ramsey numbers

Graphs and Algorithms International audience The irredundant Ramsey number s - s(m, n) [upper domination Ramsey number u - u(m, n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order s [u, respectively], it holds that IR(B) \textgreater= m or IR(R) \textgreater= n [Gamma (B) \textgreater= m or Gamma(R) \textgreater= n, respectively], where Gamma and IR denote respectively the upper domination number and the irredundance number of a graph. Furthermore, the mixed irredundant Ramsey number t = t(m, n) [mixed domination Ramsey number v = v(m, n), respectively] is the smallest natural number t [v, respectively] such that in any red-blue edge colouring (R, B) of the complete graph of order t [v, respectively], it holds that IR(B) \textgreater= m or beta(R) \textgreater= n [Gamma(B) \textgreater= m or beta(R) \textgreater= n, respectively], where beta denotes the independent domination number of a graph. These four classes of non-classical Ramsey numbers have previously been studied in the literature. In this paper we introduce a new Ramsey number w = w(m, n), called the irredundant-domination Ramsey number, which is the smallest natural number w such that in any red-blue edge colouring (R, B) of the complete graph of order w, it holds that IR(B) \textgreater= m or Gamma(R) \textgreater= n. A computer search is employed to determine complete sets of avoidance colourings of small order for these five classes of nonclassical Ramsey numbers. In the process the fifteen previously unknown Ramsey numbers t(4, 4) = 14, t(6, 3) = 17, u(4, 4) = 13, v(4, 3) = 8, v(4, 4) = 14, v(5, 3) = 13, v(6, 3) = 17, w(3, 3) = 6, w(3, 4) = w(4, 3) = 8, w(4, 4) = 13, w(3, 5) = w(5, 3) = 12 and w(3, 6) = w(6, 3) = 15 are established.

Vizing-like Conjecture for the Upper Domination of Cartesian Products of Graphs – The Proof

In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs $G$ and $H$ the upper domination number of the Cartesian product $G \,\square \, H$ is at least the product of their upper domination numbers, in symbols: $\Gamma(G \,\square \, H)\ge \Gamma(G)\Gamma(H).$