Gallai and anti-Gallai graphs
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چکیده
This chapter deals with two graph classes the Gallai graphs and the anti-Gallai graphs. \Ve construct infinitely many pairs of graphs G and H such that f( G) = r( H). The existence of a finite family offorbidden subgraphs for the Gallai graphs and the ant.i-Gallai graphs to be H-frce, for any fillit.e graph H is proved and the forbidden subgraph characterizations of G for which the Gallai graphs and the anti-Ga.llai graphs are cographs, split. graphs and threshold graphs are discussed in detail. If G is Cl connected co graph without a universal vertex then r( G) is a cograph if and only if G = (pK2)c. The relationships between t.he radius, t.he diameter and t.he chromatic number of Cl graph and it.s Gallai (r:l.llt.i-Gallai) graph a.rc a 180 st ndicd in detail. --.. -. -------.. -~Some result.s of this chapter are included ill the following pap~'r.
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A survey of the studies on Gallai and anti-Gallai graphs
The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph L(G). The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in L(G). Attributed to Gallai (1967), the study ...
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