arXiv subjects

Nevena Pivač

Publications and source records attributed to Nevena Pivač.

10 recordsLinked to original sources

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↗

arXiv2024

A tame vs. feral dichotomy for graph classes excluding an induced minor or induced topological minor

A minimal separator in a graph is an inclusion-minimal set of vertices that separates some fixed pair of nonadjacent vertices. A graph class is said to be tame if there exists a polynomial upper bound for the number of minimal separators of every graph in the class, and feral if it contains arbitrarily large graphs with exponentially many minimal separators. Building on recent works of Gartland and Lokshtanov [SODA 2023] and Gajarský, Jaffke, Lima, Novotná, Pilipczuk, Rzążewski, and Souza [arXiv, 2022], we show that every graph class defined by a single forbidden induced minor or induced topological minor is either tame or feral, and classify the two cases. This leads to new graph classes in which Maximum Weight Independent Set and many other problems are solvable in polynomial time. We complement the classification results with polynomial-time recognition algorithms for the maximal tame graph classes appearing in the obtained classifications.

math.CO↗

arXiv2023

Bisimplicial separators

A minimal separator of a graph $G$ is a set $S \subseteq V(G)$ such that there exist vertices $a,b \in V(G) \setminus S$ with the property that $S$ separates $a$ from $b$ in $G$, but no proper subset of $S$ does. For an integer $k\ge 0$, we say that a minimal separator is $k$-simplicial if it can be covered by $k$ cliques and denote by $\mathcal{G}_k$ the class of all graphs in which each minimal separator is $k$-simplicial. We show that for each $k \geq 0$, the class $\mathcal{G}_k$ is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for $\mathcal{G}_k$. We also give a complete list of minimal forbidden induced minors for $\mathcal{G}_2$. Next, we show that, for $k \geq 1$, every nonnull graph in $\mathcal{G}_k$ has a $k$-simplicial vertex, i.e., a vertex whose neighborhood is a union of $k$ cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in $\mathcal{G}_2$. Further, we show that, for $k \geq 3$, it is NP-hard to recognize graphs in $\mathcal{G}_k$; the time complexity of recognizing graphs in $\mathcal{G}_2$ is unknown. We also show that the Maximum Clique problem is NP-hard for graphs in $\mathcal{G}_3$. Finally, we prove a decomposition theorem for diamond-free graphs in $\mathcal{G}_2$ (where the diamond is the graph obtained from $K_4$ by deleting one edge), and we use this theorem to obtain polynomial-time algorithms for the Vertex Coloring and recognition problems for diamond-free graphs in $\mathcal{G}_2$, and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

math.CO↗

arXiv2023

Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition

A graph is well-covered if all its maximal independent sets have the same cardinality. This well studied concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph $G$, a real-valued vertex weight function $w$ is said to be a well-covered weighting of $G$ if all its maximal independent sets are of the same weight. The set of all well-covered weightings of a graph $G$ forms a vector space over the field of real numbers, called the well-covered vector space of $G$. Since the problem of recognizing well-covered graphs is $\mathsf{co}$-$\mathsf{NP}$-complete, the problem of computing the well-covered vector space of a given graph is $\mathsf{co}$-$\mathsf{NP}$-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graph. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies that well-covered fork-free graphs can be recognized in polynomial time and also generalizes some known results on cographs.

math.CO↗

arXiv2022

Fair allocation of indivisible items with conflict graphs

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Namely, we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both Partition and Independent Set. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).

cs.DM↗

arXiv2022

Allocation of Indivisible Items with Individual Preference Graphs

This paper studies the allocation of indivisible items to agents, when each agent's preferences are expressed by means of a directed acyclic graph. The vertices of each preference graph represent the subset of items approved of by the respective agent. An arc $(a,b)$ in such a graph means that the respective agent prefers item $a$ over item $b$. We introduce a new measure of dissatisfaction of an agent by counting the number of non-assigned items which are approved of by the agent and for which no more preferred item is allocated to the agent. Considering two problem variants, we seek an allocation of the items to the agents in a way that minimizes (i) the total dissatisfaction over all agents or (ii) the maximum dissatisfaction among the agents. For both optimization problems we study the status of computational complexity and obtain NP-hardness results as well as polynomial algorithms with respect to natural underlying graph structures, such as stars, trees, paths, and matchings. We also analyze the parameterized complexity of the two problems with respect to various parameters related to the number of agents, the dissatisfaction threshold, the vertex degrees of the preference graphs, and the treewidth.

cs.MA↗

arXiv2021

Graphs where Search Methods are Indistinguishable

Graph searching is one of the simplest and most widely used tools in graph algorithms. Every graph search method is defined using some particular selection rule, and the analysis of the corresponding vertex orderings can aid greatly in devising algorithms, writing proofs of correctness, or recognition of various graph families. We study graphs where the sets of vertex orderings produced by two different search methods coincide. We characterise such graph families for ten pairs from the best-known set of graph searches: Breadth First Search (BFS), Depth First Search (DFS), Lexicographic Breadth First Search (LexBFS) and Lexicographic Depth First Search (LexDFS), and Maximal Neighborhood Search (MNS).

cs.DM↗

arXiv2019

On the End-Vertex Problem of Graph Searches

End vertices of graph searches can exhibit strong structural properties and are crucial for many graph algorithms. The problem of deciding whether a given vertex of a graph is an end-vertex of a particular search was first introduced by Corneil, Köhler and Lanlignel in 2010. There they showed that this problem is in fact NP-complete for LBFS on weakly chordal graphs. A similar result for BFS was obtained by Charbit, Habib and Mamcarz in 2014. Here, we prove that the end-vertex problem is NP-complete for MNS on weakly chordal graphs and for MCS on general graphs. Moreover, building on previous results, we show that this problem is linear for various searches on split and unit interval graphs.

cs.DM↗

arXiv2019

Minimal separators in graph classes defined by small forbidden induced subgraphs

Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on sets of forbidden induced subgraphs with at most four vertices and obtain an almost complete dichotomy, leaving open only two cases.

math.CO↗

arXiv2018

Recognizing Graph Search Trees

Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show NP-completeness. We complement our results by providing linear time algorithms for these searches on split graphs.

cs.DM↗