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A graph is O k -free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) O k -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of O 2 -free graphs without K 2 , 3 as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in O k -free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set , Minimum Vertex Cover , Minimum Dominating Set , Minimum Coloring) can be solved in polynomial time in sparse O k -free graphs, and that deciding the O k -freeness of sparse graphs is polynomial time solvable. [ABSTRACT FROM AUTHOR]

Let G be an intersection graph of n geometric objects in the plane. We show that a maximum matching in G can be found in O (ρ 3 ω / 2 n ω / 2) time with high probability, where ρ is the density of the geometric objects and ω > 2 is a constant such that n × n matrices can be multiplied in O (n ω) time. The same result holds for any subgraph of G, as long as a geometric representation is at hand. For this, we combine algebraic methods, namely computing the rank of a matrix via Gaussian elimination, with the fact that geometric intersection graphs have small separators. We also show that in many interesting cases, the maximum matching problem in a general geometric intersection graph can be reduced to the case of bounded density. In particular, a maximum matching in the intersection graph of any family of translates of a convex object in the plane can be found in O (n ω / 2) time with high probability, and a maximum matching in the intersection graph of a family of planar disks with radii in [ 1 , Ψ ] can be found in O (Ψ 6 log 11 n + Ψ 12 ω n ω / 2) time with high probability. [ABSTRACT FROM AUTHOR]

For any small positive real ε and integer t > 1 ε , we build a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2 (1 − ε) t. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least 2 − ε. Except for the one in oriented twin-width, these lower bounds are essentially tight. [ABSTRACT FROM AUTHOR]

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ , i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f (k) n 2 k - 1 -time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b -Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K t , t -free graphs for b -Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest. [ABSTRACT FROM AUTHOR]

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connectedk-Dominating Set and Totalk-Dominating Set (albeit with a worse upper bound on the twin-width). The k-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to k-Independent Dominating Set, k-Path, k-Induced Path, k-Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connectedk-Vertex Cover and Capacitatedk-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik–Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate O (k 1.5) vertex kernel for Connectedk-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1. [ABSTRACT FROM AUTHOR]

We investigate the parameterized complexity in a and b of determining whether a graph G has a subset of a vertices and b edges whose removal disconnects G, or disconnects two prescribed vertices s , t ∈ V (G) . [ABSTRACT FROM AUTHOR]

A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time f (pw) n o (pw) on n-vertex graphs of constant degree, with pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter tl + Δ , where tl is the tree-length and Δ the maximum-degree of the input graph. [ABSTRACT FROM AUTHOR]

We give essentially tight bounds for, ν (d , k) , the maximum number of distinct neighbourhoods on a set X of k vertices in a graph with twin-width at most d. Using the celebrated Marcus–Tardos theorem, two independent works (Bonnet et al., 2022; Przybyszewski, 2022) have shown the upper bound ν (d , k) ⩽ exp (exp (O (d))) k , with a double-exponential dependence in the twin-width. The work of Gajarsky et al. (2022), using the framework of local types, implies the existence of a single-exponential bound (without explicitly stating such a bound). We give such an explicit bound, and prove that it is essentially tight. Indeed, we give a short self-contained proof that for every d and k ν (d , k) ⩽ (d + 2) 2 d + 1 k = 2 d + log d + Θ (1) k , and build a bipartite graph implying ν (d , k) ⩾ 2 d + log d + Θ (1) k , in the regime when k is large enough compared to d. [ABSTRACT FROM AUTHOR]