The edge C₄graph of some graph classes

Abstract

The edge C 4 graph of a graph G, E 4 (G) is a graph whose vertices are the edges of G and two vertices in E 4 (G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C 4. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C 4 graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E 4 (G). It is shown that for any graph G without isolated vertices, there exists a super graph H such that C(H) = G and C(E 4 (H)) = E 4 (G). Also we give forbidden subgraph characterizations for E 4 (G) being a threshold graph, block graph, geodetic graph and weakly geodetic graph.