# The edge C₄ graph of some graph classes

• Volume: 30, Issue: 3, page 365-375
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Manju K. Menon, and A. Vijayakumar. "The edge C₄ graph of some graph classes." Discussiones Mathematicae Graph Theory 30.3 (2010): 365-375. <http://eudml.org/doc/270807>.

@article{ManjuK2010,
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.},
author = {Manju K. Menon, A. Vijayakumar},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {edge C₄ graph; threshold graph; block graph; geodetic graph; weakly geodetic graph; edge graph},
language = {eng},
number = {3},
pages = {365-375},
title = {The edge C₄ graph of some graph classes},
url = {http://eudml.org/doc/270807},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Manju K. Menon
AU - A. Vijayakumar
TI - The edge C₄ graph of some graph classes
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 3
SP - 365
EP - 375
AB - 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.
LA - eng
KW - edge C₄ graph; threshold graph; block graph; geodetic graph; weakly geodetic graph; edge graph
UR - http://eudml.org/doc/270807
ER -

## References

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1. [1] A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes (SIAM, 1999). Zbl0919.05001
2. [2] V. Chvátal and P.L. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math. 1 (1997) 145-162.
3. [3] D.G. Corneil, Y. Perl and I.K. Stewart, A linear recognition algorithm for cographs, SIAM J. Comput. 14 (1985) 926-934, doi: 10.1137/0214065. Zbl0575.68065
4. [4] S. Foldes and P.L. Hammer, The Dilworth number of a graph, Ann. Discrete Math. 2 (1978) 211-219, doi: 10.1016/S0167-5060(08)70334-0. Zbl0389.05048
5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1988). Zbl0890.05002
6. [6] E. Howorka, On metric properties of certain clique graphs, J. Combin. Theory (B) 27 (1979) 67-74, doi: 10.1016/0095-8956(79)90069-8. Zbl0337.05138
7. [7] D.C. Kay and G. Chartrand, A characterization of certain Ptolemic graphs, Canad. J. Math. 17 (1965) 342-346, doi: 10.4153/CJM-1965-034-0. Zbl0139.17301
8. [8] M. Knor, L. Niepel and L. Soltes, Centers in line graphs, Math. Slovaca 43 (1993) 11-20.
9. [9] M.K. Menon and A. Vijayakumar, The edge C₄ graph of a graph, in: Proc. International Conference on Discrete Math. Ramanujan Math. Soc. Lect. Notes Ser. 7 (2008) 245-248. Zbl1202.05116
10. [10] O. Ore, Theory of Graphs, Amer. Math. Soc. Coll. Publ. 38, (Providence R.I, 1962).
11. [11] E. Prisner, Graph Dynamics (Longman, 1995). Zbl0848.05001
12. [12] S.B. Rao, A. Lakshmanan and A. Vijayakumar, Cographic and global cographic domination number of a graph, Ars Combin. (to appear). Zbl1249.05295
13. [13] D.B. West, Introduction to Graph Theory (Prentice Hall of India, 2003).

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