Geodesic
The edge C₄ graph of some graph classes
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 3, pp. 365-375.

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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 
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Menon, Manju; Vijayakumar, A. The edge C₄ graph of some graph classes. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 3, pp. 365-375. http://geodesic.mathdoc.fr/item/DMGT_2010_30_3_a0/

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