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Maria Chudnovsky

Publications and source records attributed to Maria Chudnovsky.

At least 109 records · Page 6Linked to original sources
arXiv2020

List-three-coloring $ P_t $-free graphs with no induced 1-subdivision of $ K_{1,s} $

Let $s$ and $t$ be positive integers. We use $P_t$ to denote the path with $t$ vertices and $K_{1,s}$ to denote the complete bipartite graph with parts of size $1$ and $s$ respectively. The one-subdivision of $K_{1,s}$ is obtained by replacing every edge $\{u,v\}$ of $K_{1,s}$ by two edges $\{u,w\}$ and $\{v,w\}$ with a new vertex $w$. In this paper, we give a polynomial-time algorithm for the list-three-coloring problem restricted to the class of $P_t$-free graph with no induced 1-subdivision of $K_{1,s}$.

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arXiv2020

Pure pairs. II. Excluding all subdivisions of a graph

We prove for every graph H there exists a>0 such that, for every graph G with at least two vertices, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least a|G| neighbours, or there are two disjoint sets A,B of at least a|G| vertices such that no edge joins A and B. It follows that for every graph H, there exists c>0 such that for every graph G, if no induced subgraph of G or its complement is a subdivision of H, then G has a clique or stable set of cardinality at least |G|^c. This is related to the Erdos-Hajnal conjecture.

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arXiv2020

Even-hole-free graphs still have bisimplicial vertices

A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is {\em bisimplicial} if the set of its neighbours is the union of two cliques. In an earlier paper \cite{bisimplicial}, Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial vertex, but we have recently been shown that the "proof" has a serious error. Here we give a proof using a different method.

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arXiv2020

Finding a shortest odd hole

An odd hole in a graph is a induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t, there is a polynomial time algorithm to test whether a graph contains an odd hole of length at least t. In this paper, we give an algorithm that finds a shortest odd hole, if one exists.

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arXiv2020

Finding large $H$-colorable subgraphs in hereditary graph classes

We study the \textsc{Max Partial $H$-Coloring} problem: given a graph $G$, find the largest induced subgraph of $G$ that admits a homomorphism into $H$, where $H$ is a fixed pattern graph without loops. Note that when $H$ is a complete graph on $k$ vertices, the problem reduces to finding the largest induced $k$-colorable subgraph, which for $k=2$ is equivalent (by complementation) to \textsc{Odd Cycle Transversal}. We prove that for every fixed pattern graph $H$ without loops, \textsc{Max Partial $H$-Coloring} can be solved: $\bullet$ in $\{P_5,F\}$-free graphs in polynomial time, whenever $F$ is a threshold graph; $\bullet$ in $\{P_5,\textrm{bull}\}$-free graphs in polynomial time; $\bullet$ in $P_5$-free graphs in time $n^{\mathcal{O}(ω(G))}$; $\bullet$ in $\{P_6,\textrm{1-subdivided claw}\}$-free graphs in time $n^{\mathcal{O}(ω(G)^3)}$. Here, $n$ is the number of vertices of the input graph $G$ and $ω(G)$ is the maximum size of a clique in~$G$. Furthermore, combining the mentioned algorithms for $P_5$-free and for $\{P_6,\textrm{1-subdivided claw}\}$-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for \textsc{Max Partial $H$-Coloring} in these classes of graphs. Finally, we show that even a restricted variant of \textsc{Max Partial $H$-Coloring} is $\mathsf{NP}$-hard in the considered subclasses of $P_5$-free graphs, if we allow loops on $H$.

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arXiv2020

Induced subgraphs of bounded treewidth and the container method

A hole in a graph is an induced cycle of length at least 4. A hole is long if its length is at least 5. By $P_t$ we denote a path on $t$ vertices. In this paper we give polynomial-time algorithms for the following problems: the Maximum Weight Independent Set problem in long-hole-free graphs, and the Feedback Vertex Set problem in $P_5$-free graphs. Each of the above results resolves a corresponding long-standing open problem. An extended $C_5$ is a five-vertex hole with an additional vertex adjacent to one or two consecutive vertices of the hole. Let $\mathcal{C}$ be the class of graphs excluding an extended $C_5$ and holes of length at least $6$ as induced subgraphs; $\mathcal{C}$ contains all long-hole-free graphs and all $P_5$-free graphs. We show that, given an $n$-vertex graph $G \in \mathcal{C}$ with vertex weights and an integer $k$, one can in time $n^{\Oh(k)}$ find a maximum-weight induced subgraph of $G$ of treewidth less than $k$. This implies both aforementioned results.

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arXiv2020

New Examples of Minimal Non-Strongly-Perfect Graphs

A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide several new minimal non-strongly-perfect graphs.

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arXiv2020

Cooperative colorings of trees and of bipartite graphs

Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class $\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $Ω(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$ and $Ω(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$ is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.

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arXiv2020

Better 3-coloring algorithms: excluding a triangle and a seven vertex path

We present an algorithm to color a graph $G$ with no triangle and no induced $7$-vertex path (i.e., a $\{P_7,C_3\}$-free graph), where every vertex is assigned a list of possible colors which is a subset of $\{1,2,3\}$. While this is a special case of the problem solved in [Combinatorica 38(4):779--801, 2018], that does not require the absence of triangles, the algorithm here is both faster and conceptually simpler. The complexity of the algorithm is $O(|V(G)|^5(|V(G)|+|E(G)|))$, and if $G$ is bipartite, it improves to $O(|V(G)|^2(|V(G)|+|E(G)|))$. Moreover, we prove that there are finitely many minimal obstructions to list 3-coloring $\{P_t,C_3\}$-free graphs if and only if $t \leq 7$. This implies the existence of a polynomial time certifying algorithm for list 3-coloring in $\{P_7,C_3\}$-free graphs. We furthermore determine other cases of $t, \ell$, and $k$ such that the family of minimal obstructions to list $k$-coloring in $\{P_t,C_{\ell}\}$-free graphs is finite.

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arXiv2020

On the Maximum Weight Independent Set Problem in graphs without induced cycles of length at least five

A hole in a graph is an induced cycle of length at least $4$, and an antihole is the complement of an induced cycle of length at least $4$. A hole or antihole is long if its length is at least $5$. For an integer $k$, the $k$-prism is the graph consisting of two cliques of size $k$ joined by a matching. The complexity of Maximum (Weight) Independent Set (MWIS) in long-hole-free graphs remains an important open problem. In this paper we give a polynomial time algorithm to solve MWIS in long-hole-free graphs with no $k$-prism (for any fixed integer $k$), and a subexponential algorithm for MWIS in long-hole-free graphs in general. As a special case this gives a polynomial time algorithm to find a maximum weight clique in perfect graphs with no long antihole, and no hole of length $6$. The algorithms use the framework of minimal chordal completions and potential maximal cliques.

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arXiv2019

Maximum independent sets in (pyramid, even hole)-free graphs

A \emph{hole} in a graph is an induced cycle with at least 4 vertices. A graph is \emph{even-hole-free} if it does not contain a hole on an even number of vertices. A \emph{pyramid} is a graph made of three chordless paths $P_1 = a \dots b_1$, $P_2 = a \dots b_2$, $P_3 = a \dots b_3$ of length at least~1, two of which have length at least 2, vertex-disjoint except at $a$, and such that $b_1b_2b_3$ is a triangle and no edges exist between the paths except those of the triangle and the three edges incident with $a$. We give a polynomial time algorithm to compute a maximum weighted independent set in a even-hole-free graph that contains no pyramid as an induced subgraph. Our result is based on a decomposition theorem and on bounding the number of minimal separators. All our results hold for a slightly larger class of graphs, the class of (square, prism, pyramid, theta, even wheel)-free graphs.

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arXiv2019

Subdivided Claws and the Clique-Stable Set Separation Property

Let $\mathcal{C}$ be a class of graphs closed under taking induced subgraphs. We say that $\mathcal{C}$ has the {\em clique-stable set separation property} if there exists $c \in \mathbb{N}$ such that for every graph $G \in \mathcal{C}$ there is a collection $\mathcal{P}$ of partitions $(X,Y)$ of the vertex set of $G$ with $|\mathcal{P}| \leq |V(G)|^c$ and with the following property: if $K$ is a clique of $G$, and $S$ is a stable set of $G$, and $K \cap S =\emptyset$, then there is $(X,Y) \in \mathcal{P}$ with $K \subseteq X$ and $S \subseteq Y$. In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by Göös in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant $c$ exists. In this paper we define two infinite families $\mathcal{S}, \mathcal{K}$ of graphs and show that for every $S \in \mathcal{S}$ and $K \in \mathcal{K}$, the class of graphs with no induced subgraph isomorphic to $S$ or $K$ has the clique-stable set separation property.

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arXiv2019

Small families under subdivision

Let $H$ be a graph with maximum degree $d$, and let $d'\ge 0$. We show that for some $c>0$ depending on $H,d'$, and all integers $n\ge 0$, there are at most $c^n$ unlabelled simple $d$-connected $n$-vertex graphs with maximum degree at most $d'$ that do not contain $H$ as a subdivision. On the other hand, the number of unlabelled simple $(d-1)$-connected $n$-vertex graphs with minimum degree $d$ and maximum degree at most $d+1$ that do not contain $K_{d+1}$ as a subdivision is superexponential in $n$.

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arXiv2019

Induced equators in flag spheres

We propose a combinatorial approach to the following strengthening of Gal's conjecture: $γ(Δ)\ge γ(E)$ coefficientwise, where $Δ$ is a flag homology sphere and $E\subseteq Δ$ an induced homology sphere of codimension $1$. We provide partial evidence in favor of this approach, and prove a nontrivial nonlinear inequality that follows from the above conjecture, for boundary complexes of flag $d$-polytopes: $h_1(Δ) h_i(Δ) \ge (d-i+1)h_{i-1}(Δ) + (i+1) h_{i+1}(Δ)$ for all $0\le i\le d$.

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arXiv2019

Avoidable Vertices and Edges in Graphs

A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as \textit{OCF}-vertices), simplicial paths, and their common generalization avoidable paths, introduced here. We present a general conjecture on the existence of avoidable paths. If true, the conjecture would imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices, and a result due to Chvátal, Sritharan, and Rusu from 2002 the existence of simplicial paths. In turn, both of these results generalize Dirac's classical result on the existence of simplicial vertices in chordal graphs. We prove that every graph with an edge has an avoidable edge, which settles the first open case of the conjecture. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the proved cases of the conjecture have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle.

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arXiv2019

Detecting an odd hole

A hole in a graph G is an induced cycle of length at least four; an antihole is a hole in the complement of G. In 2005, Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic showed that it is possible to test in polynomial time whether a graph contains an odd hole or antihole (and thus whether G is perfect). However, the complexity of testing for odd holes has remained open. Indeed, it seemed quite likely that testing for an odd hole was NP-complete: for instance, Bienstock showed that testing if a graph has an odd hole containing a given vertex is NP-complete. In this paper we resolve the question, by giving a polynomial-time algorithm to test whether a graph contains an odd hole. This also gives a new and considerably simpler polynomial-time algorithm that tests for perfection.

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arXiv2018

Disjoint paths in unions of tournaments

Given $k$ pairs of vertices $(s_i,t_i)\;(1\le i\le k)$ of a digraph $G$, how can we test whether there exist vertex-disjoint directed paths from $s_i$ to $t_i$ for $1\le i\le k$? This is NP-complete in general digraphs, even for $k = 2$, but in an earlier paper we proved that for all fixed $k$, there is a polynomial-time algorithm to solve the problem if $G$ is a tournament (or more generally, a semicomplete digraph). Here we prove that for all fixed $k$ there is a polynomial-time algorithm to solve the problem when $V(G)$ is partitioned into a bounded number of sets each inducing a semicomplete digraph (and we are given the partition).

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