Towards a theory of forcibly hereditary P-graphic sequences

Abstract

Let P be a hereditary invariant property of graphs. In this paper it is shown that, in principle, the set of all forcibly P-graphic sequences can be characterized using a theorem of Fulkerson, Hoffman and McAndrew and a partial order ≪ on any given set of graphic sequences A defined as: for two elements π₁, π₂, say that π₁ ≪ π₂, if there exists a graph G with degree sequence π₂ having a graph H with degree sequence π₁ as an induced subgraph. Let P be a hereditary invariant property of graphs and A(P) be the set of all degree sequence of the minimal forbidden graphs for the property P, and M(P) be the set of all minimal elements in A(P) under the partial order ≪ defined above. Conjecture: For any hereditary property P, the set M(P) is finite. We verify this conjecture for several hereditary properties P which include (1) P=Perfect-graphic, (2) P=Chromatic number of the graph ≤k, where k is a fixed positive integer; and (3) P=Planarity.