DOMINATION IN SOME GRAPHS
Project no. 20266 · Status: Kapanmış · 2018
Summary
Domination is an important parameter in graph theory, which has many applications in the analysis of communication networks, facility planning problems, electrical networks and social sciences. For this reason, many types of domination have been defined until today. In this study, semitotal domination parameter, which is a type of the domination parameter and defined by Goddard and his friends in 2014 is discussed. If each vertex in $S\subseteq V(G)$ dominating set of the graph $G$ is within 2-distance from the another vertex of $S$, $S$ is called a semitotal dominating set. The semitotal domination number of the graph $G$, denoted by $\gamma_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. A semitotal dominating set of $G$ of cardinality $\gamma_{t2}(G)$ is called a $\gamma_{t2}(G)$-set or simply a $\gamma_{t2}$-set. Here, the semitotal domination number of some important trees not in the literature was calculated. The semitotal domination number has been studied in graph operations such as corona, join, modular product and lexicographic product. Further, the exact values of the semitotal total domination number of three types of Harary graphs with high connectivity values have been obtained. Finally, semitotal bondage number, which is a new measure of vulnerability, is defined. General results have been found about this new bondage parameter. The semitotal bondage number has been calculated in basic graph families as path, cycle, wheel, star, complete bipartite graph.