Non-separating cliques, asteroidal number and leafage. The minimal 4-asteroidal split graphs

Let G be a connected chordal graph. The relation between the number of non-separating cliques of G, denoted by nsc(G), and the size of a largest asteroidal set of G, denoted by an(G), is studied. We show that an(G) is equal to the maximum nsc(H) taken over all induced connected subgraphs H of G. As a result, we provide a subclass of chordal graphs whose asteroidal number equals the leafage. We prove that the given class contains all the minimal k-asteroidal chordal graphs. Finally, we present the family of minimal 4-asteroidal split graphs.