arXiv subjects

Martin Milanič

Publications and source records attributed to Martin Milanič.

At least 19 recordsLinked to original sources

arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $α$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

math.CO↗

arXiv2026

Sandwich Monotonicity and Recognition of Weighted Graph Classes

Edge-weighted graphs play an important role in the theory of Robinsonian matrices and similarity theory, particularly via the concept of level graphs, that is, graphs obtained from an edge-weighted graph by removing all sufficiently light edges. This suggests a natural way of associating to any class $\mathcal{G}$ of unweighted graphs a corresponding class of edge-weighted graphs, namely by requiring that all level graphs belong to $\mathcal{G}$. We show that for weighted graphs $G=(V,E)$ with weights from $\{1,\dots,|E|\}$ we can decide in linear time whether all level graphs are split, threshold, or chain graphs using special edge elimination orderings. We obtain these results by introducing the notion of degree sandwich monotone graph classes. A graph class $\mathcal{G}$ is sandwich monotone if every edge set which may be removed from a graph in $\mathcal{G}$ without leaving the class also contains a single edge that can be safely removed. Furthermore, if we require the safe edge to fulfill a certain degree property, then $\mathcal{G}$ is called degree sandwich monotone. We present necessary and sufficient conditions for the existence of a linear-time recognition algorithm for any weighted graph class whose corresponding unweighted class is degree sandwich monotone and contains all edgeless graphs.

cs.DM↗

arXiv2026

On $\{k\}$-Roman graphs: complexity of recognition and the case of split graphs

For a positive integer $k$, a $\{k\}$-Roman dominating function of a graph $G = (V,E)$ is a function $f\colon V \rightarrow \{0,1,\ldots,k\}$ satisfying $\sum_{u\in N(v)} f(u) \geq k$ for each vertex $v\in V$ with $f (v) = 0$. Every graph $G$ satisfies $γ_{\{Rk\}}(G) \leq kγ(G)$, where $γ(G)$ is the domination number of $G$ and $γ_{\{Rk\}}(G)$ denotes the $\{k\}$-Roman domination number of $G$, that is, the minimum value of $\sum_{u\in V(G)} f(u)$ over all $\{k\}$-Roman dominating functions of $G$. In this work we study graphs for which the equality is reached, called \emph{$\{k\}$-Roman graphs}. This extends the concept of $\{k\}$-Roman trees studied by Wang et al.~in 2021 to general graphs. We prove that for every $k\geq 2$, the problem of recognizing \hbox{$\{k\}$-Roman} graphs is \textsf{NP}-hard, even for split graphs. For ${k\geq 3}$, we give an alternative proof by generalizing several known results on domination in middle graphs to the hypergraph setting. Finally, we characterize the \kr property within two specific subclasses of split graphs: suns and their complements.

math.CO↗

arXiv2026

Excluding paths and bicliques

Classes of graphs excluding a path and a biclique as induced subgraphs are extensively studied in the literature. One of the key structural results for such graphs is a Ramsey-type result due to Galvin, Rival, and Sands (1982), establishing the existence of a function $f$ bounding the maximum length of a path in terms of clique number $ω$. We improve the best known bound on $f$ to a function that is a singly exponential in $ω^c$, for some constant $c$, which we show is best possible, up to optimizing $c$. Our approach also has consequences for treedepth. In particular, we show that, for graphs excluding a path and a biclique as induced subgraphs, treedepth is bounded by a polynomial function of clique number. In turn, this result implies that every hereditary graph class that admits a function bounding treedepth of graphs in the class in terms of clique number, admits a polynomial such function. This gives a treedepth analogue of a recent result on pathwidth due to Hajebi (2025).

math.CO↗

arXiv2026

Clique-width and induced topological minors

A $P_4$ is a chordless path on four vertices. A diamond is a graph obtained from a clique of size four by removing one edge of the clique. A paw is a graph obtained from a clique of size four by removing two adjacent edges of the clique. We prove that for a graph $H$, the class of graphs with no induced subdivision of $H$ has bounded clique-width if and only if $H$ is an induced subgraph of $P_4$, the paw, or the diamond. This answers a~question of Dabrowski, Johnson, and Paulusma.

cs.DM↗

arXiv2026

Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique

We show that for every positive integer ${t \geq 2}$ there exists an integer $s$ such that every graph that contains no induced subgraph isomorphic to either the $6$-vertex path or the $(2,t)$-biclique, the complete bipartite graph $K_{2,t}$, has tree-independence number at most $s$. This result makes partial progress on a conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht.

math.CO↗

arXiv2026

Linear colorings of graphs

Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that treedepth could be bounded from above by twice the linear chromatic number. In this paper we investigate the properties of linear chromatic number and provide improved bounds in several graph classes.

math.CO↗

arXiv2026

Minimal toughness in subclasses of weakly chordal graphs

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in which case the toughness is defined to be infinite). A graph is said to be minimally tough if deleting any edge decreases the toughness. It is an open question whether there exists a minimally tough non-complete chordal graph with toughness exceeding $1$. We initiate the study of minimally tough graphs in the larger class of weakly chordal graphs. We obtain complete classifications of minimally tough graphs in the following subclasses of weakly chordal graphs: co-chordal graphs whose complement has diameter at least $3$, net-free co-chordal graphs, complements of forests, $P_4$-free graphs, and complete multipartite graphs. Our approach leads to simple proofs of two results on minimally tough graphs due to Dallard, Fernández, Katona, Milanič, and Varga.

math.CO↗

arXiv2026

Tree decompositions whose trees are subgraphs: An application of Simon's factorization

We show that every connected graph $G$ has a tree decomposition indexed by a tree $T$ such that $T$ is a subgraph of $G$ and the width of the tree decomposition is bounded from above by a function of the pathwidth of $G$. This answers a question of Blanco, Cook, Hatzel, Hilaire, Illingworth, and McCarty (2024), who proved that it is not possible to have such a tree decomposition whose width is bounded by a function of the treewidth of $G$. The proof relies on Simon's Factorization Theorem for finite semigroups, a tool that has already been applied successfully in various areas of graph theory and combinatorics in recent years. Our application is particularly simple and can serve as a good introduction to this technique.

math.CO↗

arXiv2026

Young domination on Hamming rectangles

We introduce a family of domination-type problems in Cartesian products of two graphs. The framework captures several well-studied topics, including variants of bootstrap percolation, line growth, distance domination, and target set selection. We focus on Cartesian products of two complete graphs and formulate the notion of Young domination number in terms of a growth rule determined by a Young diagram; this number is the smallest cardinality of an initial set that covers the entire vertex set in a prescribed number $L$ of iterations of the rule. We compute the Young domination number with $L=1$ for several natural cases, including $k$-domination for Cartesian products of two complete graphs of the same order, thereby proving a conjecture from 2009 due to Burchett, Lane, and Lachniet. We show that the case of $L=1$ of Young domination is equivalent to computing bipartite Turán numbers for families of double stars, yielding implications of our results in extremal graph theory. For arbitrary fixed $L$, we devise constant-factor approximation algorithms for the problem. Our approach is based on a variety of techniques, including duality between Young diagrams, algebraic formulations, explicit constructions, and dynamic programming.

math.CO↗

arXiv2025

Awesome graph parameters

For a graph $G$, we denote by $α(G)$ the size of a maximum independent set and by $ω(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $α$ and $ω$, respectively. One of them studies $α$-variants of graph parameters, such as $α$-treewidth or $α$-degeneracy. The second line deals with graph classes where some parameters are bounded by a function of $ω(G)$. A famous example of this type is the family of $χ$-bounded classes, where the chromatic number $χ(G)$ is bounded by a function of $ω(G)$. A Ramsey-type argument implies that if the $α$-variant of a graph parameter $ρ$ is bounded by a constant in a class $\mathcal{G}$, then $ρ$ is bounded by a function of $ω$ in $\mathcal{G}$. If the reverse implication also holds, we say that $ρ$ is awesome. Otherwise, we say that $ρ$ is awful. In the present paper, we identify a number of awesome and awful graph parameters, derive some algorithmic applications of awesomeness, and propose a number of open problems related to these notions.

math.CO↗

arXiv2025

Dominated balanced separators in wheel-induced-minor-free graphs

Gartland and Lokshtanov conjectured that every graph that excludes some planar graph as an induced minor has a balanced separator, that is, a separator whose deletion leaves every component with no more than half of the vertices of the graph, which is dominated by a bounded number of vertices. We confirm this conjecture for excluding any fixed wheel, that is, a cycle together with a universal vertex, as an induced minor.

math.CO↗

arXiv2025

Graph Classes Closed under Self-intersection

A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a disjoint union of trees with exactly one vertex of degree 3 and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard algorithmic problems admit polynomial-time algorithms. These algorithmic dichotomies, however, do not extend to the universe of all hereditary classes, which are classes closed under taking induced subgraphs. This leads to the natural question of whether we can extend known algorithmic dichotomies for monotone classes to larger families of hereditary classes. We give an affirmative answer to this question by considering the family of hereditary graph classes that are closed under self-intersection, which is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. We use our characterization to give a complete dichotomy of Maximum Independent Set, and its weighted variant for self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. This generalizes several known results on Maximum Independent Set. We also use it to obtain dichotomies for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions. Similarly, we obtain dichotomies for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions, and for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.

math.CO↗

arXiv2025

Induced matching treewidth and tree-independence number, revisited

We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition $\mathcal{T}$ of a graph $G$: for tree-independence number, the measure is the maximum size of an independent set in $G$ included in some bag of $\mathcal{T}$, while for the induced matching treewidth, the measure is the maximum size of an induced matching in $G$ such that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. While the induced matching treewidth of any graph is bounded from above by its tree-independence number, the family of complete bipartite graphs shows that small induced matching treewidth does not imply small tree-independence number. On the other hand, Abrishami, Briański, Czyżewska, McCarty, Milanič, Rzążewski, and Walczak~[SIAM Journal on Discrete Mathematics, 2025] showed that, if a fixed biclique $K_{t,t}$ is excluded as an induced subgraph, then the tree-independence number is bounded from above by some function of the induced matching treewidth. The function resulting from their proof is exponential even for fixed $t$, as it relies on multiple applications of Ramsey's theorem. In this note we show, using the Kövári-Sós-Turán theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.

cs.DM↗

arXiv2025

Closing paths to cycles in symmetric graphs

It was shown by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2022 that every induced $2$-edge path in a vertex-transitive graph closes to an induced cycle. Similar results were obtained for 3-edge paths closing to cycles in edge-transitive graphs, where the cycle can be assumed to be induced if the path is induced. Motivated by these results, we consider the following problem: For a given class of graphs, determine all integers $\ell\geq 0$ such that for every graph in the class, every path of length at most $\ell$ closes to a cycle. We also consider the variant of the problem for induced paths closing to induced cycles. We completely solve these problems for the classes of (finite) vertex-transitive graphs, edge-transitive graphs, and edge-transitive graphs that are not stars. For all but one case of a negative answer, we provide infinite families of connected counterexamples.

math.CO↗

arXiv2025

Computing Tree Decompositions with Small Independence Number

The independence number of a tree decomposition is the maximum of the independence numbers of the subgraphs induced by its bags. The tree-independence number of a graph is the minimum independence number of a tree decomposition of it. Several NP-hard graph problems, like maximum weight independent set, can be solved in time n^{O(k)} if the input n-vertex graph is given together with a tree decomposition of independence number k. Yolov, in [SODA 2018], gave an algorithm that, given an n-vertex graph G and an integer k, in time n^{O(k^3)} either constructs a tree decomposition of G whose independence number is O(k^3) or correctly reports that the tree-independence number of G is larger than k. In this paper, we first give an algorithm for computing the tree-independence number with a better approximation ratio and running time and then prove that our algorithm is, in some sense, the best one can hope for. More precisely, our algorithm runs in time 2^{O(k^2)} n^{O(k)} and either outputs a tree decomposition of G with independence number at most $8k$, or determines that the tree-independence number of G is larger than k. This implies 2^{O(k^2)} n^{O(k)}-time algorithms for various problems, like maximum weight independent set, parameterized by the tree-independence number k without needing the decomposition as an input. Assuming Gap-ETH, an n^{Ω(k)} factor in the running time is unavoidable for any approximation algorithm for the tree-independence number. Our second result is that the exact computation of the tree-independence number is para-NP-hard: We show that for every constant k \ge 4 it is NP-hard to decide if a given graph has the tree-independence number at most k.

cs.DS↗

arXiv2025

Induced Minor Models. I. Structural Properties and Algorithmic Consequences

A graph $H$ is said to be an induced minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. Equivalently, $H$ is an induced minor of $G$ if there exists an induced minor model of $H$ in $G$, that is, a collection of pairwise disjoint subsets of vertices of $G$ labeled by the vertices of $H$, each inducing a connected subgraph in $G$, such that two vertices of $H$ are adjacent if and only if there is an edge in $G$ between the corresponding subsets. In this paper, we investigate structural properties of induced minor models, including bounds on treewidth and chromatic number of the subgraphs induced by minimal induced minor models. It is known that for some graphs $H$, testing whether a given graph $G$ contains $H$ as an induced minor is an NP-complete problem. Nevertheless, as algorithmic applications of our structural results, we make use of recent developments regarding tree-independence number to show that if $H$ is the $4$-wheel, the $5$-vertex complete graph minus an edge, or a complete bipartite graph $K_{2,q}$, then there is a polynomial-time algorithm to find in a given graph $G$ an induced minor model of $H$ in $G$, if there is one. We also develop an alternative polynomial-time algorithm for recognizing graphs that do not contain $K_{2,3}$ as an induced minor, which revolves around the idea of detecting the induced subgraphs whose presence is forced when the input graph contains $K_{2,3}$ as an induced minor, using the so-called shortest path detector. It turns out that all these induced subgraphs are Truemper configurations.

math.CO↗

arXiv2025

Allocation of Indivisible Items with a Common Preference Graph: Minimizing Total Dissatisfaction

Allocating indivisible items among a set of agents is a frequently studied discrete optimization problem. In the setting considered in this work, the agents' preferences over the items are assumed to be identical. We consider a very recent measure for the overall quality of an allocation which does not rely on numerical valuations of the items. Instead, it captures the agents' opinion by a directed acyclic preference graph with vertices representing items. An arc $(a,b)$ in such a graph means that the agents prefer item $a$ over item $b$. For a given allocation of items the dissatisfaction of an agent is defined as the number of items which the agent does not receive and for which no more preferred item is given to the agent. Our goal is to find an efficient allocation of the items to the agents such that the total dissatisfaction over all agents is minimized. We explore the dichotomy between NP-hard and polynomially solvable instances, depending on properties of the underlying preference graph. While the problem is NP-hard already for three agents even on very restricted graph classes, it is polynomially solvable for two agents on general preference graphs. For an arbitrary number of agents, we derive polynomial-time algorithms for relevant restrictions of the underlying undirected graph. These are trees and, among the graphs of treewidth two, series-parallel graphs and cactus graphs.

cs.GT↗