A local reductive elimination for the fill-in of graphs

Abstract

Denote by $\text{G}$ the complementary graph of a graph G. $\text{F ⊆ E(}\text{G}\text{)}$ 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 $\text{G − v − E}{}_{\mathrm{o}}\text{, F⌣ E}{}_{\mathrm{o}}$ is a minimum fill-in of G, where $\text{E}{}_{\mathrm{o}}\text{= {xy ϵ E(}\text{G}\text{) | x, y ϵ N(v)}}$. 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.