Discussiones Mathematicae Graph Theory 30(3) (2010) 365-375
doi: 10.7151/dmgt.1499

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THE EDGE C4 GRAPH OF SOME GRAPH CLASSES

Manju K. Menon  and  A. Vijayakumar

Department of Mathematics
Cochin University of Science and Technology
Cochin-682022, India
e-mail: manjumenonk@gmail.com
e-mail: vijay@cusat.ac.in

Abstract

The edge C4 graph of a graph G, E4(G) is a graph whose vertices are the edges of G and two vertices in E4(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C4. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C4 graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E4(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(E4(H)) = E4(G). Also we give forbidden subgraph characterizations for E4(G) being a threshold graph, block graph, geodetic graph and weakly geodetic graph.

Keywords: edge C4 graph, threshold graph, block graph, geodetic graph, weakly geodetic graph.

2010 Mathematics Subject Classification: 05C99.

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Received 5 September 2008
Revised 26 March 2009
Accepted 16 July 2009