arXiv subjects
Publications and source records attributed to M. Dettlaff.
A graph $G$ is $α$-excellent if every vertex of $G$ is contained in some maximum independent set of $G$. In this paper, we characterize $α$-excellent bipartite graphs, $α$-excellent unicyclic graphs, $α$-excellent simplicial graphs, $α$-excellent chordal graphs, $α$-excellent block graphs, and we show that every generalized Petersen graph is $α$-excellent.
The neighbourhood of a vertex $v$ of a graph $G$ is the set $N(v)$ of all vertices adjacent to $v$ in $G$. For $D\subseteq V(G)$ we define $\overline{D}=V(G)\setminus D$. A set $D\subseteq V(G)$ is called a super dominating set if for every vertex $u\in \overline{D}$, there exists $v\in D$ such that $N(v)\cap \overline{D}=\{u\}$. The super domination number of $G$ is the minimum cardinality among all super dominating sets in $G$. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard.