arXiv subjects

Tomasz Krawczyk

Publications and source records attributed to Tomasz Krawczyk.

At least 19 recordsLinked to original sources

arXiv2025

A Note on the Complexity of Defensive Domination

In a graph G, a k-attack A is any set of at most k vertices and l-defense D is a set of at most l vertices. We say that defense D counters attack A if each a in A can be matched to a distinct defender d in D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and l given on input, if there exists an l-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs. The defensive domination problem is naturally in the complexity class $\Sigma^P_2$. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note we prove that the defensive domination problem is $\Sigma^P_2$-complete. We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also $\Sigma^P_2$-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.

cs.CC↗

arXiv2024

On the structure of normalized models of circular-arc graphs -- Hsu's approach revisited

Circular-arc graphs are the intersection graphs of arcs of a circle. The main result of this work describes the structure of all \emph{normalized intersection models} of circular-arc graphs. Normalized models of a circular-arc graph reflect the neighborhood relation between its vertices and can be seen as its canonical representations; in particular, any intersection model can be made normalized by possibly extending some of its arcs. We~devise a data-structure, called \emph{PQM-tree}, that maintains the set of all normalized models of a circular-arc graph. We show that the PQM-tree of a circular-arc graph can be computed in linear time. Finally, basing on PQM-trees, we provide a linear-time algorithm for the canonization and the isomorphism problem for circular-arc graphs. We describe the structure of the normalized models of circular-arc graphs using an approach proposed by Hsu~[\emph{SIAM J. Comput. 24(3), 411--439, (1995)}]. In the aforementioned work, Hsu claimed the construction of decomposition trees representing the set of all normalized intersection models of circular-arc graphs and an $\mathcal{O}(nm)$ time isomorphism algorithm for this class of graphs. However, the counterexample given in~[\emph{Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013}] shows that Hsu's isomorphism algorithm is incorrect. Also, in a companion paper we show that the decomposition trees proposed by Hsu are not constructed correctly; in particular, we showed that there are circular-arc graphs whose all normalized models do not follow the description given by Hsu.

cs.DS↗

arXiv2024

Comments on "$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs"

In the work [$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs, SIAM J. Comput. 24(3), 411--439, (1995)], Wen-Lian Hsu claims three results concerning the class of circular-arc graphs: - the design of so-called \emph{decomposition trees} that represent the structure of all normalized intersection models of circular-arc graphs, - an $\mathcal{O}(m\cdot n)$ recognition algorithm for circular-arc graphs, - an $\mathcal{O}(m\cdot n)$ isomorphism algorithm for circular-arc graphs. In [Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013] Curtis, Lin, McConnell, Nussbaum, Soulignac, Spinrad, and Szwarcfiter showed that Hsu's isomorphism algorithm is incorrect. In this note, we show that the other two results -- namely, the construction of decomposition trees and the recognition algorithm -- are also flawed.

cs.DS↗

arXiv2024

Characterization of Circular-arc Graphs: III. Chordal Graphs

We identify all minimal chordal graphs that are not circular-arc graphs, thereby resolving one of ``the main open problems'' concerning the structures of circular-arc graphs as posed by Dur{\'{a}}n, Grippo, and Safe in 2011. The problem had been attempted even earlier, and previous efforts have yielded partial results, particularly for claw-free graphs and graphs with an independence number of at most four. The answers turn out to have very simple structures: all the nontrivial ones belong to a single family. Our findings are based on a structural study of McConnell's flipping, which transforms circular-arc graphs into interval graphs with certain representation patterns.

math.CO↗

arXiv2024

Characterization of Circular-arc Graphs: II. McConnell Flipping

McConnell [FOCS 2001] presented a flipping transformation from circular-arc graphs to interval graphs with certain patterns of representations. Beyond its algorithmic implications, this transformation is instrumental in identifying all minimal graphs that are not circular-arc graphs. We conduct a structural study of this transformation, and for $C_{4}$-free graphs, we achieve a complete characterization of these patterns. This characterization allows us, among other things, to identify all minimal chordal graphs that are not circular-arc graphs in a companion paper.

math.CO↗

arXiv2024

Circular-arc graphs and the Helly property

In this paper we investigate some problems related to the Helly properties of circular-arc graphs, which are defined as intersection graphs of arcs of a fixed circle. As such, circular-arc graphs are among the simplest classes of intersection graphs whose models might not satisfy the Helly property. In particular, some cliques of a circular-arc graph might be Helly in some but not all arc intersection models of the graph. Our first result is an alternative proof of a theorem by Lin and Szwarcfiter which asserts that for every circular-arc graph $G$ either every normalized model of $G$ satisfies the Helly property or no normalized model of $G$ satisfies this property. Further, we study the Helly properties of a single clique of a circular-arc graph $G$. We divide the cliques of $G$ into three types: a clique $C$ of $G$ is always-Helly/always-non-Helly/ambiguous if $C$ is Helly in every/no/(some but not all) normalized model of $G$. We provide a combinatorial description for the cliques of each type, and based on it, we devise a polynomial time algorithm which determines the type of a given clique. Finally, we study the Helly Cliques problem, in which we are given an $n$-vertex circular-arc graph $G$ and some of its cliques $C_1, \ldots, C_k$ and we ask if there is an arc intersection model of $G$ in which all the cliques $C_1, \ldots, C_k$ satisfy the Helly property. We show that: (1) the Helly Cliques problem admits a $2^{O(k\log{k})}n^{O(1)}$-time algorithm (that is, it is FPT when parametrized by the number of cliques given in the input), (2) assuming Exponential Time Hypothesis (ETH), the Helly Cliques problem cannot be solved in time $2^{o(k)}n^{O(1)}$, (3) the Helly Cliques problem admits a polynomial kernel of size $O(k^6)$. All our results use a data structure, called a PQM-tree, which maintains all normalized models of a circular-arc graph $G$.

cs.DS↗

arXiv2024

Characterization of Chordal Circular-arc Graphs: I. Split Graphs

The most elusive problem around the class of circular-arc graphs is identifying all minimal graphs that are not in this class. The main obstacle is the lack of a systematic way of enumerating these minimal graphs. McConnell [FOCS 2001] presented a transformation from circular-arc graphs to interval graphs with certain patterns of representations. We fully characterize these interval patterns for circular-arc graphs that are split graphs, thereby building a connection between minimal split graphs that are not circular-arc graphs and minimal non-interval graphs. This connection enables us to identify all minimal split graphs that are not circular-arc graphs. As a byproduct, we develop a linear-time certifying recognition algorithm for circular-arc graphs when the input is a split graph.

math.CO↗

arXiv2022

Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs

In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph $H$, the class of $H$-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of $H$. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of $H$-graphs for different graphs $H$. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborn\'ık, and Zeman showed, for every fixed tree $T$, a polynomial-time algorithm recognizing $T$-graphs. Tucker showed a polynomial time algorithm recognizing $K_3$-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed that recognition of $H$-graphs is $NP$-hard if $H$ contains two different cycles sharing an edge. The main two results of this work narrow the gap between the $NP$-hard and $P$ cases of $H$-graphs recognition. First, we show that recognition of $H$-graphs is $NP$-hard when $H$ contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing $L$-graphs, where $L$ is a graph containing a cycle and an edge attached to it ($L$-graphs are called lollipop graphs). Our work leaves open the recognition problems of $M$-graphs for every unicyclic graph $M$ different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows. Firstly, the recognition of $M$-graphs, where $M$ is a fixed unicyclic graph, admits a polynomial time algorithm if we restrict the input to graphs containing particular holes (hence recognition of $M$-graphs is probably most difficult for chordal graphs). Secondly, the recognition of medusa graphs, which are defined as the union of $M$-graphs, where $M$ runs over all unicyclic graphs, is $NP$-complete.

cs.DS↗

arXiv2021

Grounded L-graphs are polynomially $χ$-bounded

A grounded L-graph is the intersection graph of a collection of "L" shapes whose topmost points belong to a common horizontal line. We prove that every grounded L-graph with clique number $ω$ has chromatic number at most $17ω^4$. This improves the doubly-exponential bound of McGuinness and generalizes the recent result that the class of circle graphs is polynomially $χ$-bounded. We also survey $χ$-boundedness problems for grounded geometric intersection graphs and give a high-level overview of recent techniques to obtain polynomial bounds.

math.CO↗

arXiv2021

Colouring polygon visibility graphs and their generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $ω$ has chromatic number at most $3\cdot 4^{ω-1}$. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time.

math.CO↗

arXiv2020

Vertex deletion into bipartite permutation graphs

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time $O(9^k\cdot n^9)$, and also give a polynomial-time 9-approximation algorithm.

cs.DS↗

arXiv2020

On-line partitioning of width w posets into w^O(log log w) chains

An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of a presented poset.

cs.DS↗

arXiv2019

Testing isomorphism of circular-arc graphs -- Hsu's approach revisited

Circular-arc graphs are intersection graphs of arcs on the circle. The aim of our work is to present a polynomial time algorithm testing whether two circular-arc graphs are isomorphic. To accomplish our task we construct decomposition trees, which are the structures representing all normalized intersection models of circular-arc graphs. Normalized models reflect the neighbourhood relation in circular-arc graphs and can be seen as their canonical representations; in particular, every intersection model can be easily transformed into a normalized one. Our work adapts and appropriately extends the previous work on the similar topic done by Hsu [\emph{SIAM J. Comput. 24(3), 411--439, (1995)}]. In his work, Hsu developed decomposition trees representing all normalized models of circular-arc graphs. However due to the counterexample given in [\emph{Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013}], his decomposition trees can not be used by algorithms testing isomorphism of circular-arc graphs.

cs.DS↗

arXiv2018

The Slow-coloring Game on Sparse Graphs: $k$-Degenerate, Planar, and Outerplanar

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar.

math.CO↗

arXiv2018

An easy subexponential bound for online chain partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.

cs.DS↗

arXiv2017

Defective 3-Paintability of Planar Graphs

A $d$-defective $k$-painting game on a graph $G$ is played by two players: Lister and Painter. Initially, each vertex is uncolored and has $k$ tokens. In each round, Lister marks a chosen set $M$ of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset $X$ of $M$ which induce a subgraph $G[X]$ of maximum degree at most $d$. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that $G$ is $d$-defective $k$-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.

math.CO↗

arXiv2016

On-line approach to off-line coloring problems on graphs with geometric representations

The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that on-line graph coloring problems give rise to classes of game graphs with a natural geometric interpretation. We use this concept to estimate the chromatic number of graphs with geometric representations by finding, for appropriate simpler graphs, on-line coloring algorithms using few colors or proving that no such algorithms exist. We derive upper and lower bounds on the maximum chromatic number that rectangle overlap graphs, subtree overlap graphs, and interval filament graphs (all of which generalize interval overlap graphs) can have when their clique number is bounded. The bounds are absolute for interval filament graphs and asymptotic of the form $(\log\log n)^{f(ω)}$ for rectangle and subtree overlap graphs, where $f(ω)$ is a polynomial function of the clique number and $n$ is the number of vertices. In particular, we provide the first construction of geometric intersection graphs with bounded clique number and with chromatic number asymptotically greater than $\log\log n$. We also introduce a concept of $K_k$-free colorings and show that for some geometric representations, $K_3$-free chromatic number can be bounded in terms of clique number although the ordinary ($K_2$-free) chromatic number cannot. Such a result for segment intersection graphs would imply a well-known conjecture that $k$-quasi-planar geometric graphs have linearly many edges.

cs.DS↗

arXiv2016

The Partial Visibility Representation Extension Problem

For a graph $G$, a function $ψ$ is called a \emph{bar visibility representation} of $G$ when for each vertex $v \in V(G)$, $ψ(v)$ is a horizontal line segment (\emph{bar}) and $uv \in E(G)$ iff there is an unobstructed, vertical, $\varepsilon$-wide line of sight between $ψ(u)$ and $ψ(v)$. Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph $G$, a bar visibility representation $ψ$ of $G$, additionally, puts the bar $ψ(u)$ strictly below the bar $ψ(v)$ for each directed edge $(u,v)$ of $G$. We study a generalization of the recognition problem where a function $ψ'$ defined on a subset $V'$ of $V(G)$ is given and the question is whether there is a bar visibility representation $ψ$ of $G$ with $ψ(v) = ψ'(v)$ for every $v \in V'$. We show that for undirected graphs this problem together with closely related problems are \NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time.

cs.CG↗