Journal of Prime Research in Mathematics

# Forcing edge detour number of an edge detour graph

A. P. Santhakumaran
Research Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai – 627 002, India.
S. Athisayanathan
Research Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai – 627 002, India.

$$^{1}$$Corresponding Author: apskumar1953@yahoo.co.in

### Abstract

For two vertices $$u$$ and $$v$$ in a graph $$G = (V, E)$$, the detour distance $$D(u, v)$$ is the length of a longest $$u–v$$ path in $$G$$. A $$u–v$$ path of length $$D(u, v)$$ is called a $$u–v$$ detour. A set $$S ⊆ V$$ is called an edge detour set if every edge in $$G$$ lies on a detour joining a pair of vertices of $$S$$. The edge detour number $$dn_1(G)$$ of G is the minimum order of its edge detour sets and any edge detour set of order $$dn_1(G)$$ is an edge detour basis of $$G$$. A connected graph $$G$$ is called an edge detour graph if it has an edge detour set. A subset $$T$$ of an edge detour basis $$S$$ of an edge detour graph $$G$$ is called a forcing subset for $$S$$ if $$S$$ is the unique edge detour basis containing $$T$$. A forcing subset for $$S$$ of minimum cardinality is a minimum forcing subset of $$S$$. The forcing edge detour number $$fdn_1(S)$$ of $$S$$, is the minimum cardinality of a forcing subset for $$S$$. The forcing edge detour number $$fdn_1(G)$$ of $$G$$, is $$min{fdn_1(S)}$$, where the minimum is taken over all edge detour bases $$S$$ in $$G$$. The general properties satisfied by these forcing subsets are discussed and the forcing edge detour numbers of certain classes of standard edge detour graphs are determined. The parameters $$dn_1(G)$$ and $$fdn_1(G)$$ satisfy the relation $$0 ≤ fdn_1(G) ≤ dn_1(G)$$ and it is proved that for each pair $$a$$, $$b$$ of integers with $$0 ≤ a ≤ b$$ and $$b ≥ 2$$, there is an edge detour graph $$G$$ with $$fdn_1(G) = a$$ and $$dn_1(G) = b$$.

#### Keywords:

Detour, edge detour set, edge detour basis, edge detour number,
forcing edge detour number.