Geodesic
Properties of Hereditary Hypergraphs and Middle Graphs
Canadian mathematical bulletin, Tome 21 (1978) no. 4, pp. 461-468

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The middle graph of a graph G=( V, E) is the graph M(G) = (V∪E, E′), in which two vertices u, v are adjacent if either M is a vertex in V and v is an edge in E containing u, or u and v are edges in E having a vertex in common. Middle graphs have been characterized in terms of line graphs by Hamada and Yoshimura [7], who also investigated their traversability and connectivity properties. In this paper another characterization of middle graphs is presented, in which they are viewed as a class of intersection (representative) graphs of hereditary hypergraphs. Graph theoretic parameters associated with the concepts of vertex independence, dominance, and irredundance for middle graphs are discussed, and equalities relating the chromatic number of a graph to these parameters are obtained. 
Cockayne, E. J.; Hedetniemi, S. T.; Miller, D. J. Properties of Hereditary Hypergraphs and Middle Graphs. Canadian mathematical bulletin, Tome 21 (1978) no. 4, pp. 461-468. doi: 10.4153/CMB-1978-079-5
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     author = {Cockayne, E. J. and Hedetniemi, S. T. and Miller, D. J.},
     title = {Properties of {Hereditary} {Hypergraphs} and {Middle} {Graphs}},
     journal = {Canadian mathematical bulletin},
     pages = {461--468},
     year = {1978},
     volume = {21},
     number = {4},
     doi = {10.4153/CMB-1978-079-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1978-079-5/}
}
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