Algorithmic aspects of switch cographs

Maximum Weight Stable Set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time

With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Pointer network improves the attention mechanism, instead of allocating different attention to hidden states of encoder to generate context vectors, using attention as a pointer to select an element of the input sequence at every step of decoding, which solves the problem of variable dictionary size of the output sequence. Pointer net (Ptr-Net) applied to three combinatorial optimization problems, convex hull, Delaunay triangulation, and traveling salesman problem (TSP), obtains good approximate solutions. Point matching is also a special kind of combinatorial optimization problems that is to obtain the optimal corresponding references, which can be modeled by Ptr-Net. However, Ptr-Net can’t be used to solve point matching problem because it doesn’t take full advantage of the correspondences between the two point sets. We propose multi-pointer network, which draws the idea from multi-label classification, to address this limitation by pointing out a set of input elements. These applications are all based on supervised learning to approximate expected known solutions. However, high-quality labeled data is often expensive, unreliable, or simply unavailable and may be infeasible for new problem statements, making supervised learning being unpractical. Reinforcement learning, as another research hotspot in the field of machine learning, does not require labeled sample data. It interacts with the environment through trial-and-error mechanism and focuses more on learning problem-solving strategies. We introduce a framework to tackle combinatorial optimization problems using neural networks and reinforcement learning, focusing on the traveling salesman problem. We also introduce a framework, a unique combination of reinforcement learning and graph embedding network, to solve graph optimization problems, focusing on maximum cut (MAXCUT) and minimum vertex cover (MVC) problems.

In this paper, we study the class of graphs Gm,n that have the same degree sequence as two disjoint cliques Km and Kn, as well as the class G¯m,n of the complements of such graphs. The problems of finding a maximum clique and a maximum independent set are NP-hard on Gm,n. Therefore, looking for upper and lower bounds for the clique and independence numbers of such graphs is a challenging task. In this article, we obtain such bounds, as well as other related results. In particular, we consider the class of regular graphs, which are degree-equivalent to arbitrarily many identical cliques, as well as such graphs of bounded degree.

For graph $G$ and integers $a_1 \ge \cdots \ge a_r \ge 2$, we write $G \rightarrow (a_1 ,\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \in \{1, \cdots, r\}$. The vertex Folkman number $F_v(a_1 ,\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \rightarrow (a_1 ,\cdots ,a_r)^v$. It is well known that if $G \rightarrow (a_1 ,\cdots ,a_r)^v$ then $\chi(G) \geq m$, where $m = 1+ \sum_{i=1}^r (a_i - 1)$. In this paper we study such Folkman graphs $G$ with chromatic number $\chi(G)=m$, which leads to a new concept of chromatic Folkman numbers. We prove constructively some existential results, among others that for all $r,s \ge 2$ there exist $K_{s+1}$-free graphs $G$ such that $G \rightarrow (s,\cdots_r,s)^v$ and $G$ has the smallest possible chromatic number $r(s-1)+1$ with respect to this property. Among others we conjecture that for every $s \ge 2$ there exists a $K_{s+1}$-free graph $G$ on $F_v(s,s;s+1)$ vertices with $\chi(G)=2s-1$ and $G\rightarrow (s,s)^v$.

Mind the independence gap

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. In previous works, polynomial time algorithms were found for recognizing coordinated graphs within some classes of graphs. In this paper we prove that the recognition problem for coordinated graphs is NP-hard, and it is NP-complete even when restricted to the class of {gem, C4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{𝒪(√k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√k)}(n+m)^𝒪(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{𝒪(√k)}(n+m)^𝒪(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{𝒪(√klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width 𝒪(√k).

On weighted efficient total domination

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r+2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings.Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems.

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph from an infinite sequence of attacks is the eternal vertex cover number (evc) of the graph. It is known that, given a graph G and an integer k, checking whether \({\text {evc}}(G) \le k\) is NP-Hard. However, for any graph G, \({\text {mvc}}(G) \le {\text {evc}}(G) \le 2 {\text {mvc}}(G)\), where \({\text {mvc}}(G)\) is the minimum vertex cover number of G. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. Though a characterization is known for graphs for which \({\text {evc}}(G) = 2{\text {mvc}}(G)\), a characterization of graphs for which \({\text {evc}}(G) = {\text {mvc}}(G)\) remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine \({\text {evc}}(G)\) and to determine a safe strategy of guard movement in each round of the game with \({\text {evc}}(G)\) guards.

Let [Formula: see text] and [Formula: see text] respectively denote the chromatic number and clique number of a graph [Formula: see text]. A [Formula: see text] is a path on 5 vertices, a banner (paw) is the graph obtained by joining a new vertex to a single vertex of [Formula: see text] ([Formula: see text]) and a hammer is obtained by subdividing the pendant edge of a paw exactly once. Recently, [Formula: see text]-free graphs have received wide attention. In 2019, Karthick, Maffray and Pastor, gave a structural characterisation of [Formula: see text]-free graphs, which when combined with a result by Bourneuf and Thomassé (2023) implies that for a [Formula: see text]-free graph [Formula: see text], [Formula: see text]. Geißer in his thesis (2022) showed that the [Formula: see text]-binding function of the class of [Formula: see text]-free graphs is bounded by the [Formula: see text]-binding function of [Formula: see text]-free graphs. By a result of Kim (1995), the chromatic number [Formula: see text] of a [Formula: see text]-free graph [Formula: see text] has order of magnitude [Formula: see text]. Recently, Song and Xu (2024) proved that every ([Formula: see text], [Formula: see text], banner, hammer)-free graph [Formula: see text] is [Formula: see text]-colorable. This motivates us to study the subclasses of ([Formula: see text], banner)-free graphs. We prove that for any ([Formula: see text], banner, [Formula: see text])-free graph [Formula: see text] where [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, the bound is tight for [Formula: see text].

The question of characterizing graphs H such that the Vertex Cover problem is solvable in polynomial time in the class of H-free graphs is notoriously difficult and still widely open. We completely solve the corresponding question for a distance-based generalization of vertex cover called distance-k vertex cover, for any positive integer k. In this problem the task is to determine, given a graph G and an integer \(\ell \), whether G contains a set of at most \(\ell \) vertices such that each edge of G is at distance at most k from a vertex in the set. We show that for all \(k \ge 1\) and all graphs H, the distance-k vertex cover problem is solvable in polynomial time in the class of H-free graphs if H is an induced subgraph of \(P_{2k+2} + sP_{\max \{k,2\}}\) for some \(s \ge 0\), and NP-complete otherwise. KeywordsDistance-k Vertex CoverH-free graphNP-completenessPolynomial-time algorithmDichotomy

S-functions are mappings from the class of finite graphs into the set of integers, such that certain formal conditions are fulfilled which are shared by the chromatic number, the vertex-connectivity, and the homomorphism-degree. The S-functions form a complete lattice (with respect to their natural partial order). The classes of graphs with values <n under some S-function are studied from a general point of view, and uncountably many S-functions are constructed. Further for every n≥5 a non-trivial base-element of (see K. WAGNER [7]) is constructed.

Fully dynamic representations of interval graphs

Counting spanning trees using modular decomposition