Minimum Weighted Coloring of Triangulated Graphs, with Application to Maximum Weight Vertex Packing and Clique Finding in Arbitrary Graphs

Efficient algorithms are known for finding a maximum weight stable set, a minimum weighted clique covering, and a maximum weight clique of a vertex-weighted triangulated graph. However, there is no comparably efficient algorithm in the literature for finding a minimum weighted vertex coloring of such a graph. This paper gives an O(] VI 2) procedure for the problem (Algorithm 1). It then extends the procedure to the problem of finding in an arbitrary graph G--(V, E) a maximal induced subgraph G(W) colorequivalent (as defined in 3) to a maximal triangulated subgraph G(T) (Algorithm 2). Finally, it uses this latter algorithm as the main ingredient of a branch-and-bound procedure for the maximum weight clique problem in an arbitrary graph. Computational experience is presented on arbitrary random graphs with up to 2,000 vertices. Key words, graph coloring, vertex packing, maximum clique finding, triangulated graphs AMS(MOS) subject classifications. 05, 90, 68