The Normal Graph Conjecture for Two Classes of Sparse Graphs

Abstract

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover $$\mathcal {Q}$$ and a stable set cover $$\mathcal {S}$$ s.t. every clique in $$\mathcal {Q}$$ intersects every stable set in $$\mathcal {S}$$ . Normal graphs can be considered as closure of perfect graphs by means of co-normal products and graph entropy. Perfect graphs have been characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al.). Korner and de Simone observed that $$C_5$$ , $$C_7$$ , and $$\overline{C}_7$$ are minimal not normal and conjectured, in analogy to the Strong Perfect Graph Theorem, that every $$(C_5, C_7, \overline{C}_7)$$ -free graph is normal (Normal Graph Conjecture, Korner and de Simone). Recently, this conjecture has been disproved by Harutyunyan, Pastor and Thomasse. However, in this paper we verify it for two classes of sparse graphs, 1-trees and cacti. In addition, we provide both linear time recognition algorithms and characterizations for the normal graphs within these two classes.