The Upper Forcing Edge-to-Vertex Geodetic Number of a Graph
Journal: International Journal of Mathematics and Soft Computing (Vol.6, No. 1)Publication Date: 2016.01.09
Authors : Venkiteshwaran Sujitha;
Page : 29-38
Keywords : edge-to-vertex geodetic number; forcing edge-to-vertex geodetic number; upper forcing edge-to-vertex geodetic number.;
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.
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