IJPAM: Volume 103, No. 1 (2015)

THE FORCING EDGE-TO-VERTEX GEODETIC
NUMBER OF A GRAPH

S. Sujitha$^1$, J. John$^2$, A. Vijayan$^3$
$^1$Department of Mathematics
Holy Cross College (Autonomous)
Nagercoil, 629004, INDIA
$^2$Department of Mathematics
Government College of Engineering
Tirunelveli, 627007, INDIA
$^3$Department of Mathematics
N.M. Christian College
Marthandam, 629165, INDIA


Abstract. For a connected graph $G=(V,E)$, a set $S \subseteq E$ is called an 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 a 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 edge-to-vertex geodetic basis of $G$. A subset $T \subseteq S$ is called a 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 forcing edge-to-vertex geodetic number of $S$, denoted by $f_{ev}(S)$, is the cardinality of a minimum forcing subset of $S$. The forcing edge-to-vertex geodetic number of $G$, denoted by $f_{ev}(G)$, is $f_{ev}(G) = min \left\{f_{ev}(S)\right\}$, where the minimum is taken over all minimum edge-to-vertex geodetic sets $S$ in $G$. Some general properties satisfied by the concept forcing edge-to-vertex geodetic number is studied. The forcing edge-to-vertex geodetic number of certain classes of graphs are determined. It is shown that for every pair $a, b$ of integers with $0 \leq a < b$, there exists a connected graph $G$ such that $f_{ev}(G) = a$ and $g_{ev}(G) = b$.

Received: April 21, 2015

AMS Subject Classification: 05C12

Key Words and Phrases: edge-to-vertex geodetic number, forcing edge-to-vertex geodetic number

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DOI: 10.12732/ijpam.v103i1.9 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 103
Issue: 1
Pages: 109 - 121


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