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 − Eo, F⌣ Eo is a minimum fill-in of G, where Eo = {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.
