The Upper Forcing Edge-to-Vertex Geodetic Number of a Graph

Abstract

For a connected graph $G=(V,E)$, a set $S \subseteq E$ is called an \textit{edge-to-vertex geodetic set} of $G$ if every vertex of $G$ is either incident with an edge of $S$ or lies on a geodesic joining some pair of edges of $S$. The minimum cardinality of an edge-to-vertex geodetic set of $G$ is $g_{ev}(G)$. Any edge-to-vertex geodetic set of cardinality $g_{ev}(G)$ is called an \emph{edge-to-vertex geodetic basis} of $G$. A subset $T \subseteq S$ is called a \emph{forcing} subset for $S$ if $S$ is the unique minimum edge-to-vertex geodetic set containing $T$. A forcing subset for $S$ of minimum cardinality is a minimum forcing subset of $S$. The \emph{forcing edge-to-vertex geodetic number} of $S$, denoted by $f_{ev}(S)$, is the cardinality of a minimum forcing subset of $S$. The \emph{upper forcing edge-to-vertex geodetic number} of $G$, denoted by $f^{+}_{ev}(G)$, is $f^{+}_{ev}(G) = max \left\{f_{ev}(S)\right\}$, where the maximum is taken over all minimum edge-to-vertex geodetic sets $S$ in $G$. It is shown that the upper forcing edge-to-vertex geodetic number lies between 0 and $g_{ev}(G)$. Also, the upper forcing edge-to-vertex geodetic number of certain classes of graphs such as cycle, tree, complete graph and complete bipartite graph are determined.