A local reductive elimination for the fill-in of graphs

Abstract

Denote by G the complementary graph of a graph G. F ⊆ E( G) is called a fill-in of G, if G + F is a chordal graph. A minimum fill-in of G is a fill-in of G with minimum cardinality. In this paper we show that, for a k-connected graph G, if v is a vertex of G with degree k, and there exists a ( k − 1)-clique of G contained in N( v), then for every minimum fill-in F of G − v − E o, F⌣ E o is a minimum fill-in of G, where E o = {xy ϵ E( G) | x, y ϵ N(v)} . Using this local reductive elimination we design a linear time algorithm to solve the minimum fill-in problem for Halin graphs. We also show that the cardinality of the minimum fill-in of a Halin graph can be denoted by a formula.