Powers of geometric intersection graphs and dispersion algorithms

Subexponential-Time Algorithms for Maximum Independent Set in $$P_t$$ P t -Free and Broom-Free Graphs

“…Scattered Set (also known under other names such as dispersion or distanced independent set [1,7,12,23,26,28]) is the natural generalization of MIS where the vertices of the solution are required to be at distance at least d from each other; the size of the largest such set will be denoted by α d (G). We can consider with d being part of the input, or assume that d ≥ 2 is a fixed constant, in which case we call the problem d-Scattered Set.…”

“…The algorithmic side of Theorem 4 is based on the combinatorial observation that the treewidth of P t -free graphs is sublinear in the number of edges, which means that standard algorithms on bounded-treewidth graphs can be invoked to solve the problem in time subexponential in the number of edges. It has not escaped our notice that this approach is completely generic and could be used for many other problems (e.g., Hamiltonian Cycle, 3-Coloring, and so on), where 1) -time algorithms are known on graphs of treewidth t. For the lowerbound part of Theorem 4, we need to examine only two cases: claw-free graphs and C t -free graphs (where C t is the cycle on t vertices); the other cases then follow immediately.…”

“…As in the proof of Theorem 6, as long as there exists a vertex in G of degree larger than Δ, we can branch on such a vertex v: in one subcase, we consider independent sets not containing v (and thus delete v from G), in the other subcase, we consider independent sets containing v (and thus delete N (v) from G). Such a branching step can be conducted in polynomial time, and fits in the second subcase of max in (1). Thus, we can assume henceforth that the maximum degree of G is at most Δ.…”

“…Designing an algorithm based on deep results, Lokshtanov et al [20] finally proved that MIS is polynomial-time solvable on P 5 -free graphs. More recently, a quasipolynomial (n log O (1) n -time) algorithm was found for P 6 -free graphs [19] and finally a polynomial-time algorithm for P 6 -free graphs was announced [14]. For P 7 -free graphs, a partial result is known: MWIS is polynomialtime solvable if we additionally exclude a triangle [8].…”

“…We explore MIS and some variants on H -free graphs from the viewpoint of subexponential-time algorithms in this work. That is, instead of aiming for algorithms with running time n O (1) on n-vertex graphs, we ask if 2 o(n) algorithms are possible. Very recently, Brause [9] and independently the conference version of this paper [4] observed that the subexponential algorithm of Randerath and Schiermeyer [25] can be generalized to arbitrary fixed t ≥ 5 with running time roughly 2 O(n 1−1/t ) .…”