Edge Complete \((p,2)\) Semigraphs

Abstract

A semigraph \(G\) is an ordered pair \((V,X)\) where \(V\) is a non-empty set whose elements are called vertices of \(G\) and \(X\) is a set of \(n\)-tuples (\(n > 2\)), called edges of \(G\), of distinct vertices satisfying the following conditions:

i) any edge \((v_1, v_2, \ldots, v_n)\) of \(G\) is the same as its reverse \((v_n, v_{n-1}, \ldots, v_1)\),and

ii) any two edges have at most one vertex in common.

Two edges are adjacent if they have a common vertex. \(G\) is edge complete if any two edges in \(G\) are adjacent. In this paper, we enumerate the non-isomorphic edge complete \((p,2)\)semigraphs.