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.

On the Paired-Domination Subdivision Number of Trees

A paired-dominating set of a graph G without isolated vertices is a dominating set of vertices whose induced subgraph has perfect matching. The minimum cardinality of a paired-dominating set of G is called the paired-domination number γpr(G) of G. The paired-domination subdivision number sdγpr(G) of G is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. Here, we show that, for each tree T≠P5 of order n≥3 and each edge e∉E(T), sdγpr(T)+sdγpr(T+e)≤n+2.

Graphical Analysis of Covering and Paired Domination in the Environment of Neutrosophic Information

Neutrosophic graph (NG) is a powerful tool in graph theory, which is capable of modeling many real-life problems with uncertainty due to unclear, varying, and indeterminate information. Meanwhile, the fuzzy graphs (FGs) and intuitionistic fuzzy graphs (IFGs) may not handle these problems as efficiently as NGs. It is difficult to model uncertainty due to imprecise information and vagueness in real-world scenarios. Many real-life optimization problems are modeled and solved using the well-known fuzzy graph theory. The concepts of covering, matching, and paired domination play a major role in theoretical and applied neutrosophic environments of graph theory. Henceforth, the current study covers this void by introducing the notions of covering, matching, and paired domination in single-valued neutrosophic graph (SVNG) using the strong edges. Also, many attention-grabbing properties of these concepts are studied. Moreover, the strong covering number, strong matching number, and the strong paired domination number of complete SVNG, complete single-valued neutrosophic cycle (SVNC), and complete bipartite SVNG are worked out along with their fascinating properties.

On the Paired-Domination Subdivision Number of a Graph

In order to increase the paired-domination number of a graph G, the minimum number of edges that must be subdivided (where each edge in G can be subdivided no more than once) is called the paired-domination subdivision number sdγpr(G) of G. It is well known that sdγpr(G+e) can be smaller or larger than sdγpr(G) for some edge e∉E(G). In this note, we show that, if G is an isolated-free graph different from mK2, then, for every edge e∉E(G), sdγpr(G+e)≤sdγpr(G)+2Δ(G).

A Note on the Paired-Domination Subdivision Number of Trees

For a graph G with no isolated vertex, let γpr(G) and sdγpr(G) denote the paired-domination and paired-domination subdivision numbers, respectively. In this note, we show that if T is a tree of order n≥4 different from a healthy spider (subdivided star), then sdγpr(T)≤min{γpr(T)2+1,n2}, improving the (n−1)-upper bound that was recently proven.

Upper paired domination in graphs

<abstract><p>A set $ PD\subseteq V(G) $ in a graph $ G $ is a paired dominating set if every vertex $ v\notin PD $ is adjacent to a vertex in $ PD $ and the subgraph induced by $ PD $ contains a perfect matching. A paired dominating set $ PD $ of $ G $ is minimal if there is no proper subset $ PD'\subset PD $ which is a paired dominating set of $ G $. A minimal paired dominating set of maximum cardinality is called an upper paired dominating set, denoted by $ \Gamma_{pr}(G) $-set. Denote by $ Upper $-$ PDS $ the problem of computing a $ \Gamma_{pr}(G) $-set for a given graph $ G $. Michael et al. showed the APX-completeness of $ Upper $-$ PDS $ for bipartite graphs with $ \Delta = 4 $ ^{[<xref ref-type="bibr" rid="b11">11</xref>]}. In this paper, we show that $ Upper $-$ PDS $ is APX-complete for bipartite graphs with $ \Delta = 3 $.</p></abstract>