A Linear Algorithm for Maximum Weight Cliques in Proper Circular Arc Graphs
Binay Kumar Bhattacharya, Pavol Hell, Jing Huang · SIAM Journal on Discrete Mathematics · 1996
We present an $O(n)$algorithm to find a maximum clique in a proper circular arc graph. We assume that the input graph is represented by a sorted simple family of circular arcs or by an equivalent representation. In Deng, Hell, and Huang [SIAM J. Comput., 25 (1996), pp. 390–403], we gave an $O(m + n)$ algorithm to find such a representation for a proper circular arc graph given by its adjacency lists. As an application we also give an $O(n)$ algorithm for q-coloring proper circular arc graphs for a fixed q. (Such an algorithm was first given by Teng and Tucker.) Finally we indicate how our algorithm can be modified to find a maximum weight clique in a weighted graph, also in time $O(n)$.