Computing and Counting Longest Paths on Circular-Arc Graphs in Polynomial Time

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O ( n 4 ) ) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G ′ of G admits a path P ′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O ( n ) . As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O ( n 4 ) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight.