Counting independent sets in cocomparability graphs

Abstract

We show that the number of independent sets in cocomparability graphs can be counted in linear time, as can counting cliques in comparability graphs. By contrast, counting cliques in cocomparabilty graphs and counting independent sets in comparabilty graphs are #Pcomplete. We extend these results to counting maximal cliques and independent sets. We also consider the fixed-parameter versions of counting cliques and independent sets of given size k. Finally, we combine the results to show that both counting cliques and independent sets in permutation graphs are in linear time.

Counting independent sets in graphs with bounded bipartite pathwidth

“…Counting independent sets in graphs, determining , is known to be #P‐complete in general [41], and in various restricted cases [28, 45]. Exact counting in polynomial time is known only for some restricted graph classes, for example, [22]. Even approximate counting is NP‐hard in general, and is unlikely to be in polynomial time for bipartite graphs [20].…”

Total Independent Set Numbers on Catacondensed Polyomino Systems

“…Moreover, its connection to the hard core model in statistical physics is well established [2]. Total independent set number is a graph invariant that has been studied extensively, we can refer to [1][2][3][4][5][6] and references therein. Polyominoes have a long and rich history, we convey for the origin polyominoes in [7].…”

Counting independent sets in strongly orderable graphs

“…holds for other classes of graphs such as graphs of maximum degree 3 [Gre00], planar graphs [Vad01] or comparability graphs [DM19]. From the point of view of parametrized complexity, counting the number of independent sets of a given size was shown to be W [1]-hard [FG04], and hence is unlikely to have an FPT algorithm.…”

“…On the positive side, the problem has a polynomial time algorithm for several more restricted classes of graphs such as chordal graphs [OUU08], cocomparability graphs [DM19], graphs with bounded tree-width [WTZL18] or tolerance graphs [LS15], just to give a few examples. A picture representing all the known results, and the relations between the different classes can be found in Figure 1.…”

“…Similarly to independent sets, the problem was studied for several classes of graphs, and was shown to be #P-hard on chordal and chordal bipartite graphs [OUU09]. Surprisingly, while it is also hard on planar graphs [DM19], the problem of counting only perfect matchings for this class is related to Pfaffian orientations and admits a polynomial time algorithm [Kas63].…”