Coupling and Bernoullicity in random-cluster and Potts models

An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity

Dominator Coloring and CD Coloring in Almost Cluster Graphs

In this paper, we study two popular variants of Graph Coloring -- Dominator Coloring and CD Coloring. In both problems, we are given a graph $G$ and a natural number $\ell$ as input and the goal is to properly color the vertices with at most $\ell$ colors with specific constraints. In Dominator Coloring, we require for each $v \in V(G)$, a color $c$ such that $v$ dominates all vertices colored $c$. In CD Coloring, we require for each color $c$, a $v \in V(G)$ which dominates all vertices colored $c$. These problems, defined due to their applications in social and genetic networks, have been studied extensively in the last 15 years. While it is known that both problems are fixed-parameter tractable (FPT) when parameterized by $(t,\ell)$ where $t$ is the treewidth of $G$, we consider strictly structural parameterizations which naturally arise out of the problems' applications. We prove that Dominator Coloring is FPT when parameterized by the size of a graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT parameterized by CVD set size plus the number of remaining cliques. En route, we design a simpler and faster FPT algorithms when the problems are parameterized by the size of a graph's twin cover, a special CVD set. When the parameter is the size of a graph's clique modulator, we design a randomized single-exponential time algorithm for the problems. These algorithms use an inclusion-exclusion based polynomial sieving technique and add to the growing number of applications using this powerful algebraic technique.Comment: 29 pages, 3 figure

On the Approximability of the Exemplar Adjacency Number Problem for Genomes with Gene Repetitions

In this paper, we apply a measure, exemplar adjacency number, which complements and extends the well-studied breakpoint distance between two permutations, to measure the similarity between two genomes (or in general, between any two sequences drawn from the same alphabet). For two genomes and drawn from the same set of n gene families and containing gene repetitions, we consider the corresponding Exemplar Adjacency Number problem (EAN), in which we delete duplicated genes from and such that the resultant exemplar genomes (permutations) G and H have the maximum adjacency number. We obtain the following results. First, we prove that the one-sided 2-repetitive EAN problem, i.e., when one of and is given exemplar and each gene occurs in the other genome at most twice, can be linearly reduced from the Maximum Independent Set problem. This implies that EAN does not admit any -approximation algorithm, for any , unless P = NP. This hardness result also implies that EAN, parameterized by the optimal solution value, is W[1]-hard. Secondly, we show that the two-sided 2-repetitive EAN problem has an -approximation algorithm, which is tight up to a constant factor

On dd-stable locally checkable problems parameterized by mim-width

In this paper we continue the study of locally checkable problems under the framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on graphs of bounded mim-width. We study which restrictions on a locally checkable problem are necessary in order to be able to solve it efficiently on graphs of bounded mim-width. To this end, we introduce the concept of $d$-stability of a check function. The related locally checkable problems contain large classes of problems, among which we can mention, for example, LCVP problems. We give an algorithm showing that these problems are XP when parameterized by the mim-width of a given binary decomposition tree of the input graph, that is, that they can be solved in polynomial time given a binary decomposition tree of bounded mim-width. We explore the relation between $d$-stable locally checkable problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke, 2022), and show that both frameworks model the same family of problems. We include a list of concrete examples of $d$-stable locally checkable problems whose complexity on graphs of bounded mim-width was open so far

A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

Defensive Alliances in Signed Networks

The analysis of (social) networks and multi-agent systems is a central theme in Artificial Intelligence. Some line of research deals with finding groups of agents that could work together to achieve a certain goal. To this end, different notions of so-called clusters or communities have been introduced in the literature of graphs and networks. Among these, defensive alliance is a kind of quantitative group structure. However, all studies on the alliance so for have ignored one aspect that is central to the formation of alliances on a very intuitive level, assuming that the agents are preconditioned concerning their attitude towards other agents: they prefer to be in some group (alliance) together with the agents they like, so that they are happy to help each other towards their common aim, possibly then working against the agents outside of their group that they dislike. Signed networks were introduced in the psychology literature to model liking and disliking between agents, generalizing graphs in a natural way. Hence, we propose the novel notion of a defensive alliance in the context of signed networks. We then investigate several natural algorithmic questions related to this notion. These, and also combinatorial findings, connect our notion to that of correlation clustering, which is a well-established idea of finding groups of agents within a signed network. Also, we introduce a new structural parameter for signed graphs, signed neighborhood diversity snd, and exhibit a parameterized algorithm that finds a smallest defensive alliance in a signed graph