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In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n1−ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs H such as P5, P6, the claw, or the fork. We prove that for every such “possibly tractable” graph H there exists an algorithm that, given an H-free graph G and an accuracy parameter ε > 0, finds an independent set in G of cardinality within a factor of (1 – ε) of the optimum in time exponential in a polynomial of log | V(G) | and ε−1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APX-hard in H-free graphs, the problem admits a quasi-polynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.
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Square-Free Graphs with no Induced Fork
The claw is the graph $$K_{1,3}$$, and the fork is the graph obtained from the claw $$K_{1,3}$$ by subdividing one of its edges once. In this paper, we prove a structure theorem for the class of (claw, $$C_4$$)-free graphs that are not quasi-line graphs, and a structure theorem for the class of (fork, $$C_4$$)-free graphs that uses the class of (claw, $$C_4$$)-free graphs as a basic class. Finally, we show that every (fork, $$C_4$$)-free graph $$G$$ satisfies $$\chi(G)\leqslant \lceil\frac{3\omega(G)}{2}\rceil$$ via these structure theorems with some additional work on coloring basic classes.
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- Award ID(s):
- 1763817
- PAR ID:
- 10289368
- Date Published:
- Journal Name:
- The Electronic Journal of Combinatorics
- Volume:
- 28
- Issue:
- 2
- ISSN:
- 1077-8926
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract A conjecture of Alon, Krivelevich and Sudakov states that, for any graph $$F$$ , there is a constant $$c_F \gt 0$$ such that if $$G$$ is an $$F$$ -free graph of maximum degree $$\Delta$$ , then $$\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$$ . Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs $$F$$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if $$G$$ is $$K_{t,t}$$ -free, then $$\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$$ as $$\Delta \to \infty$$ . We improve this bound to $$(1+o(1)) \Delta/\log\!\Delta$$ , making the constant factor independent of $$t$$ . We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.more » « less
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For graphs G and H, we say that G is H-free if it does not contain H as an induced subgraph. Already in the early 1980s Alekseev observed that if H is connected, then the Max Weight Independent Set problem (MWIS) remains NP-hard in H-free graphs, unless H is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in H-free graphs, where H is any fixed path. If H is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw H, MWIS is polynomial-time solvable in H-free graphs of bounded degree.more » « less
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For a graph F, a graph G is F-free if it does not contain an induced subgraph isomorphic to F. For two graphs G and H, an H-coloring of G is a mapping f : V (G) → V (H) such that for every edge uv ∈ E(G) it holds that f(u)f(v) ∈ E(H). We are interested in the complexity of the problem H-Coloring, which asks for the existence of an H-coloring of an input graph G. In particular, we consider H-Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5. This problem is closely related to the well known open problem of determining the complexity of 3-Coloring of Pt-free graphs. We show that for every odd k ≥ 5 the Ck-Coloring problem, even in the precoloring-extension variant, can be solved in polynomial time in P9-free graphs. On the other hand, we prove that the extension version of Ck-Coloring is NP-complete for F-free graphs whenever some component of F is not a subgraph of a subdivided claw.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.37236/9144
