Perfectness of clustered graphs

Flavia Bonomo (1) , Denis Cornaz (2) , Tinaz Ekim (3) , Bernard Ries (2)

1 Universitad de Buenos Aires = University of Buenos Aires [Argentina]
2 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
3 AUTRES

Given a clustered graph (G,V), that is, a graph G=(V,E) together with a partition V of its vertex set, the selective coloring problem consists in choosing one vertex per cluster such that the chromatic number of the subgraph induced by the chosen vertices is minimum. This problem can be formulated as a covering problem with a 0–1 matrix M(G,V). Nevertheless, we observe that, given (G,V), it is NP-hard to check if M(G,V) is conformal (resp. perfect). We will give a sufficient condition, checkable in polynomial time, for M(G,V) to be conformal that becomes also necessary if conformality is required to be hereditary. Finally, we show that M(G,V) is perfect for every partition V if and only if G belongs to a superclass of threshold graphs defined with a complex function instead of a real one.

Mots clés: Threshold graph; Perfect matrix; Conformal matrix; Partition coloring; Selective coloring

Domaines: Informatique [cs]

Flavia Bonomo, Denis Cornaz, Tinaz Ekim, Bernard Ries. Perfectness of clustered graphs. Discrete Optimization, 2013, 10 (4), ⟨10.1016/j.disopt.2013.07.006⟩. ⟨hal-01497133⟩