Testing Superperfection of k-Trees

An interval coloring of a weighted graph with non-negative weights, maps each vertex onto an open interval on the real line with width equal to the weight of the vertex, such that adjacent vertices are mapped to disjoint intervals. The total width of an interval coloring is defined as the width of the union of the intervals. The interval chromatic number of a weighted graph is the least total width of an interval coloring. The weight of a subset of vertices is the sum of the weights of the vertices in the subset. The clique number of a weighted graph is the weight of the heaviest clique in the graph. A graph is called superperfect if, for every non-negative weight function, the clique number is equal to the interval chromatic number. A k-tree is a graph which can be recursively defined as follows. A clique with k + 1 vertices is a k-tree. Given a k-tree with n vertices, a k-tree with n + 1 vertices can be obtained by making a new vertex adjacent to all vertices of a k-clique in the k-tree. In this paper we present, for each constant k, a linear time algorithm to test if a k-tree is superperfect. We also give, for each constant k, a constant time algorithm to produce a complete characterization of superperfect k-trees. Finally we present a complete list of critical non-superperfect 2-trees. Answering a question of Golumbic ([11]), this shows the existence of triangulated graphs which are superperfect but not comparability graphs.