H-free Graphs Research Articles

Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for (P6+sP2)-free graphs for every s≥0, extending a known result for P5-free graphs (Bouquet and Picouleau, 2020).

Classifying subset feedback vertex set for H-free graphs

In the Feedback Vertex Set problem, we aim to find a small set S of vertices in a graph intersecting every cycle. The Subset Feedback Vertex Set problem requires S to intersect only those cycles that include a vertex of some specified set T. We also consider the Weighted Subset Feedback Vertex Set problem, where each vertex u has weight w(u)>0 and we ask that S has small weight. By combining known NP-hardness results with new polynomial-time results we prove full complexity dichotomies for Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set for H-free graphs, that is, graphs that do not contain a graph H as an induced subgraph.

Complexity of the (Connected) Cluster Vertex Deletion Problem on H-free Graphs

The well-known Cluster Vertex Deletion problem (cluster-vd) asks for a given graph G and an integer k whether it is possible to delete a set S of at most k vertices of G such that the resulting graph G-S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G-S$$\\end{document} is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs H for which cluster-vd on H-free graphs is polynomially solvable and for which it is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-complete. Moreover, in the NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-completeness cases, cluster-vd cannot be solved in sub-exponential time in the vertex number of the H-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of cluster-vd, the Connected Cluster Vertex Deletion problem (connected cluster-vd), in which the set S has to induce a connected subgraph of G. It turns out that connected cluster-vd admits the same complexity dichotomy for H-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on H-free graphs.

Reconfiguration of vertex colouring and forbidden induced subgraphs

The reconfiguration graph of the k-colourings, denoted Rk(G), is the graph whose vertices are the k-colourings of G and two colourings are adjacent in Rk(G) if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of Rk+1(G) for a k-colourable graph G restricted by forbidden induced subgraphs. We show that Rk+1(G) is connected for every k-colourable H-free graph G if and only if H is an induced subgraph of P4 or P3+P1. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if G is a k-colourable (2K2, C4)-free graph, then Rk+1(G) is connected with diameter at most 4n. Furthermore, we show that Rk+1(G) is connected for every k-colourable (P5, C4)-free graph G.

Hitting all maximum stable sets in P5-free graphs

We prove that every P5-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where Pt denotes the t-vertex path, and for graphs G,H, we say G is H-free if no induced subgraph of G is isomorphic to H).More generally, let us say a class C of graphs is η-bounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest cardinality of a hitting set of all maximum stable sets in G, and ω(G) is the clique number of G. Also, C is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial.We introduce η-boundedness inspired by a question of Alon (asking how large η(G) can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,•given a graph G, we have η(H)≤ω(H) for every induced subgraph H of G if and only if G is perfect;•there are graphs G with both η(G) and the girth of G arbitrarily large; and•if C is a hereditary class of graphs which is polynomially η-bounded, then C satisfies the Erdős-Hajnal conjecture. The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all H-free graphs is η-bounded if (and only if) H is a forest. Like χ-boundedness, the case where H is a star is easy to verify, and we prove two non-trivial extensions of this: H-free graphs are η-bounded if (1) H has a vertex incident with all edges of H, or (2) H can be obtained from a star by subdividing at most one edge, exactly once.Unlike χ-boundedness, the case where H is a path is surprisingly hard. Our main result mentioned at the beginning shows that P5-free graphs are η-bounded. The proof is rather involved compared to the classical “Gyárfás path” argument which establishes, for all t, the χ-boundedness of Pt-free graphs. It remains open whether Pt-free graphs are η-bounded for t≥6.It also remains open whether P5-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for P5-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of P5. We further generalize the case where H is a 1-regular graph on four vertices, showing that H-free graphs are polynomially η-bounded if H is a forest with no vertex of degree more than one and at most four vertices of degree one.

Vertex-critical [formula omitted]-free and vertex-critical (gem, co-gem)-free graphs

A graph G is k-vertex-critical if χ(G)=k but χ(G−v)<k for all v∈V(G) where χ(G) denotes the chromatic number of G. We show that there are only finitely many k-vertex-critical (P3+ℓP1)-free graphs for all k≥1 and all ℓ≥0. Together with previous results, the only graphs H for which it is unknown if there are an infinite number of k-vertex-critical H-free graphs is H=(P4+ℓP1) for all ℓ≥1. We consider a restriction on the smallest open case, and show that there are only finitely many k-vertex-critical (gem, co-gem)-free graphs for all k, where gem=P4+P1¯. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of C5. This characterization allows us to give the complete list of all k-vertex-critical (gem, co-gem)-free graphs for all k≤16.

Ramsey–Turán problems with small independence numbers

Given a graph H and a function f(n), the Ramsey–Turán number RT(n,H,f(n)) is the maximum number of edges in an n-vertex H-free graph with independence number at most f(n). For H being a small clique, many results about RT(n,H,f(n)) are known and we focus our attention on H=Ks for s≤13.By applying Szemerédi’s Regularity Lemma, the dependent random choice method and some weighted Turán-type results, we prove that these cliques have the so-called phase transitions when f(n) is around the inverse function of the off-diagonal Ramsey number of Kr versus a large clique Kn for some r≤s.

Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs

We say a class C of graphs is clean if for every positive integer t there exists a positive integer w(t) such that every graph in C with treewidth more than w(t) contains an induced subgraph isomorphic to one of the following: the complete graph Kt, the complete bipartite graph Kt,t, a subdivision of the (t×t)-wall or the line graph of a subdivision of the (t×t)-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all H-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph H) is clean if and only if H is a forest whose components are subdivided stars.Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest H as above, we show that forbidding certain connected graphs containing H as an induced subgraph (rather than H itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer η, a complete description of unavoidable connected induced subgraphs of a connected graph G containing η vertices from a suitably large given set of vertices in G. This is of independent interest, and will be used in subsequent papers in this series.

On the price of independence for vertex cover, feedback vertex set and odd cycle transversal

Let vc(G), fvs(G) and oct(G), respectively, denote the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph G. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If G has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let ivc(G), ifvs(G) or ioct(G), respectively, denote the minimum size of such a set. Similar to a recent study on the price of connectivity (Hartinger et al. EuJC 2016), we investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are bounded in terms of vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find complete classifications for vertex cover and feedback vertex set and an almost complete classification for odd cycle transversal (subject to three non-equivalent open cases). We also investigate for which graphs H the values of ivc(G), ifvs(G) and ioct(G) are equal to vc(G), fvs(G) and oct(G), respectively, when the graph G belongs to the class of H-free graphs. We find a complete classification for vertex cover and almost complete classifications for feedback vertex set (subject to one open case) and odd cycle transversal (subject to three open cases).

The maximum number of triangles in [formula omitted]-free graphs

The generalized Turán number ex(n,Ks,H) is the maximum number of complete graph Ks in an H-free graph on n vertices. Let Fk be the friendship graph consisting of k triangles. Erdős and Sós (1976) determined the value of ex(n,K3,F2). Alon and Shikhelman (2016) proved that ex(n,K3,Fk)≤(9k−15)(k+1)n. In this paper, by using a method developed by Chung and Frankl in hypergraph theory, we determine the exact value of ex(n,K3,Fk) and the extremal graph for any Fk when n≥4k3.

Strengthening Rödl's theorem

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rödl showed in the 1980s that every H-free graph has large parts that are very sparse or very dense. More precisely, let us say that a graph F on n vertices is ε-restricted if either F or its complement has maximum degree at most εn. Rödl proved that for every graph H, and every ε>0, every H-free graph G has a linear-sized set of vertices inducing an ε-restricted graph. We strengthen Rödl's result as follows: for every graph H, and all ε>0, every H-free graph can be partitioned into a bounded number of subsets inducing ε-restricted graphs.

Finding Matching Cuts in H-Free Graphs

The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r>0 such that the first variant is NP-complete for P_r-free graphs. This addresses a question of Bouquet and Picouleau (The complexity of the Perfect Matching-Cut problem. CoRR, arXiv:2011.03318, (2020)). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs.

Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Paths P^1,ldots ,P^k in a graph G=(V,E) are mutually induced if any two distinct P^i and P^j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (s_i,t_i) contains k mutually induced paths P^i such that each P^i starts from s_i and ends at t_i. This is a classical graph problem that is NP-complete even for k=2. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.

Polynomial bounds for chromatic number VI. Adding a four-vertex path

A hereditary class of graphs is χ-bounded if there is a function f such that every graph G in the class has chromatic number at most f(ω(G)), where ω(G) is the clique number of G; and the class is polynomiallyχ-bounded if f can be taken to be a polynomial. The Gyárfás–Sumner conjecture asserts that, for every forest H, the class of H-free graphs (graphs with no induced copy of H) is χ-bounded. Let us say a forest H is good if it satisfies the stronger property that the class of H-free graphs is polynomially χ-bounded.Very few forests are known to be good: for example, the goodness of the five-vertex path is open. Indeed, it is not even known that if every component of a forest H is good then H is good, and in particular, it was not known that the disjoint union of two four-vertex paths is good. Here we show the latter (with corresponding polynomial ω(G)16); and more generally, that if H is good then so is the disjoint union of H and a four-vertex path. We also prove an even more general result: if every component of H1 is good, and H2 is any path (or broom) then the class of graphs that are both H1-free and H2-free is polynomially χ-bounded.

A spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs

A graph G is called H-free, if it does not contain H as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of an H-free graph of order n? In this paper, we consider the Brualdi-Solheid-Turán type problem for non-bipartite graphs. Let Ka,b•K3 denote the graph obtained by identifying a vertex of Ka,b in the part of size b and a vertex of K3. We prove that if G is a non-bipartite graph of order n satisfying ρ(G)≥ρ(K⌈n−22⌉,⌊n−22⌋•K3), then G contains all odd cycles C2l+1 for each integer l∈[2,k] unless G≅K⌈n−22⌉,⌊n−22⌋•K3, provided that n is sufficiently large with respect to k. This resolves the problem posed by Guo, Lin and Zhao (2021).

Minimum Degree Stability of H-Free Graphs

Given an (r + 1)-chromatic graph H, the fundamental edge stability result of Erdős and Simonovits says that all n-vertex H-free graphs have at most (1 - 1/r + o(1)) binom{n}{2} edges, and any H-free graph with that many edges can be made r-partite by deleting o(n^{2}) edges. Here we consider a natural variant of this—the minimum degree stability of H-free graphs. In particular, what is the least c such that any n-vertex H-free graph with minimum degree greater than cn can be made r-partite by deleting o(n^{2}) edges? We determine this least value for all 3-chromatic H and for very many non-3-colourable H (all those in which one is commonly interested) as well as bounding it for the remainder. This extends the Andrásfai-Erdős-Sós theorem and work of Alon and Sudakov.

Rational Exponents Near Two

A longstanding conjecture of Erd˝os and Simonovits states that for every rational r between 1 and 2 there is a graph H such that the largest number of edges in an H-free graph on n vertices is Q(nr). Answering a question raised by Jiang, Jiang and Ma, we show that the conjecture holds for all rationals of the form 2􀀀a=b with b sufficiently large in terms of a.

MAX-CUT BY EXCLUDING BIPARTITE SUBGRAPHS

AbstractFor a graph G, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of G. Given a positive integer m and a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$ , as G ranges over all graphs on m edges that contain no copy of H. We prove bounds on $f(m,H)$ for some bipartite graphs H and give a bound for a conjecture of Alon et al. [‘MaxCut in H-free graphs’, Combin. Probab. Comput.14 (2005), 629–647] concerning $f(m,K_{4,s})$ .

Few induced disjoint paths for H-free graphs

Paths P1,…,Pk in a graph G=(V,E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices. For a fixed integer k, the k-Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that each Pi starts from si and ends at ti. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer k, a classical result from the literature states that even 2-Induced Disjoint Paths is NP-complete. We prove new complexity results for k-Induced Disjoint Paths if the input is restricted to H-free graphs, that is, graphs without a fixed graph H as an induced subgraph. We compare our results with a complexity dichotomy for Induced Disjoint Paths, the variant where k is part of the input.

Parameterized Inapproximability of Independent Set in H-Free Graphs

We study the Independent Set problem in H-free graphs, i.e., graphs excluding some fixed graph H as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halldórsson [SODA 1995] showed that for every \(\delta >0\) the Independent Set problem has a polynomial-time \((\frac{d-1}{2}+\delta )\)-approximation algorithm in \(K_{1,d}\)-free graphs. We extend this result by showing that \(K_{a,b}\)-free graphs admit a polynomial-time\({\mathcal {O}}(\alpha (G)^{1-1/a})\)-approximation, where \(\alpha (G)\) is the size of a maximum independent set in G. Furthermore, we complement the result of Halldórsson by showing that for some \(\gamma =\Theta (d/\log d),\) there is no polynomial-time \(\gamma \)-approximation algorithm for these graphs, unless NP = ZPP. Bonnet et al. [Algorithmica 2020] showed that Independent Set parameterized by the size k of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least 4, (2) the star \(K_{1,4}\), and (3) any tree with two vertices of degree at least 3 at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken conditions (1) and (2) that G does not contain a cycle of constant length at least 5 or \(K_{1,5}\)). First, under the ETH, there is no \(f(k) \cdot n^{o(k/\log k)}\) algorithm for any computable function f. Then, under the deterministic Gap-ETH, there is a constant \(\delta >0\) such that no \(\delta \)-approximation can be computed in \(f(k) \cdot n^{O(1)}\) time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime \(f(k) \cdot n^{o(\sqrt{k})}\). Finally, we consider the parameterization by the excluded graph H, and show that under the ETH, Independent Set has no \(n^{o(\alpha (H))}\) algorithm in H-free graphs. Also, we prove that there is no \(d/k^{o(1)}\)-approximation algorithm for \(K_{1,d}\)-free graphs with runtime \(f(d,k) \cdot n^{{\mathcal {O}}(1)}\), under the deterministic Gap-ETH.