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Kristina Vušković

Publications and source records attributed to Kristina Vušković.

At least 19 recordsLinked to original sources
arXiv2024

Tree independence number I. (Even hole, diamond, pyramid)-free graphs

The tree-independence number tree-$α$, first defined and studied by Dallard, Milanič and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so-called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass $\mathcal C$ of (even hole, diamond, pyramid)-free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that $\mathcal C$ has bounded tree-$α$. Via existing results, this yields a polynomial time algorithm for the maximum independent set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič and Štorgel that in a hereditary graph class, tree-$α$ is bounded if and only if the treewidth is bounded by a function of the clique number.

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arXiv2024

Submodular functions and perfect graphs

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.

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arXiv2023

Graphs with all holes the same length

A graph is "$\ell$-holed" if all its induced cycles of length at least four have length exactly $\ell$. We give a complete description of the $\ell$-holed graphs for each $\ell\ge 7$.

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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.

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arXiv2023

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all $k$ and $Δ$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a subdivision of the $(k\times k)$-wall or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For $t\geq 2$, a $t$-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least $t$. A $t$-pyramid is a graph consisting of a vertex $v$, a triangle $B$ disjoint from $v$ and three paths starting at $v$ and disjoint otherwise, each joining $v$ to a vertex of $B$, and each of length at least $t$. We prove that for all $k,t$ and $Δ$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a $t$-theta, or a $t$-pyramid, or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a $t$-theta for some $t\geq 2$). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every $Δ$ and subcubic subdivided caterpillar $T$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a subdivision of $T$ or the line graph of a subdivision of $T$ as an induced subgraph.

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arXiv2022

Induced subgraphs and tree decompositions V. One neighbor in a hole

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions"). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

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arXiv2022

Induced subgraphs and tree decompositions V. Small components of big vertices

Aboulker, Adler, Kim, Sintiari, and Trotignon conjectured that every graph with bounded maximum degree and large treewidth must contain, as an induced subgraph, a large subdivided wall, or the line graph of a large subdivided wall. This conjecture was recently proved by Korhonen, but the problem of identifying the obstacles to bounded treewidth in the general case (that is, without the bounded maximum degree condition) remains wide open. Examples of structures of large treewidth which avoid the "usual suspects" have been constructed by Sintiari and Trotignon, and by Davies. In this note, we aim to better isolate the features of these examples that lead to large treewidth. To this end, we prove the following result. Let $G$ be a graph, and write $γ(G)$ for the size of a largest connected component in the graph induced by $G$ on the set of vertices of degree at least 3. If $γ(G)$ is small and the treewidth of $G$ is large, then $G$ must contain a large subdivided wall or the line graph of a large subdivided wall. This result is the best possible, in the sense that the conclusion fails if we replace 3 by any larger number in the definition of $γ(G)$, as evidenced by Davies' example.

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arXiv2022

Induced subgraphs and tree decompositions I. Even-hole-free graphs of bounded degree

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph has treewidth $k$ if it can be decomposed by a sequence of noncrossing cutsets of size at most $k$ into pieces of size at most $k+1$. The study of hereditary graph classes (i.e. those closed under vertex deletion only) reveals a different picture, where cutsets that are not necessarily bounded in size (such as star cutsets, 2-joins and their generalization) are required to decompose the graph into simpler pieces that are structured but not necessarily bounded in size. A number of such decomposition theorems are known for complex hereditary graph classes, including even-hole-free graphs, perfect graphs and others. These theorems do not describe the global structure in the sense that a tree decomposition does, since the cutsets guaranteed by them are far from being noncrossing. They are also of limited use in algorithmic applications. We show that in the case of even-hole-free graphs of bounded degree the cutsets described in the previous paragraph can be partitioned into a bounded number of well-behaved collections. This allows us to prove that even-hole-free graphs with bounded degree have bounded treewidth, resolving a conjecture of Aboulker, Adler, Kim, Sintiari and Trotignon [arXiv:2008.05504]. As a consequence, it follows that many algorithmic problems can be solved in polynomial time for this class, and that even-hole-freeness is testable in the bounded degree graph model of property testing. In fact we prove our results for a larger class of graphs, namely the class of $C_4$-free odd-signable graphs with bounded degree.

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arXiv2021

Graphs with polynomially many minimal separators

We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent set problem for the class of (theta, pyramid, prism, turtle)-free graphs. Since every prism, theta, and turtle contains an even hole, this also implies a polynomial time algorithm to solve the maximum weight independent set problem for the class of (pyramid, even hole)-free graphs.

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arXiv2020

Coloring rings

A ring is a graph $R$ whose vertex set can be partitioned into $k \geq 4$ nonempty sets, $X_1, \dots, X_k$, such that for all $i \in \{1,\dots,k\}$, the set $X_i$ can be ordered as $X_i = \{u_i^1, \dots, u_i^{|X_i|}\}$ so that $X_i \subseteq N_R[u_i^{|X_i|}] \subseteq \dots \subseteq N_R[u_i^1] = X_{i-1} \cup X_i \cup X_{i+1}$. A hyperhole is a ring $R$ such that for all $i \in \{1,\dots,k\}$, $X_i$ is complete to $X_{i-1}\cup X_{i+1}$. In this paper, we prove that the chromatic number of a ring $R$ is equal to the maximum chromatic number of a hyperhole in $R$. Using this result, we give a polynomial-time coloring algorithm for rings. Rings formed one of the basic classes in a decomposition theorem for a class of graphs studied by Boncompagni, Penev, and Vušković in [Journal of Graph Theory 91 (2019), 192--246]. Using our coloring algorithm for rings, we show that graphs in this larger class can also be colored in polynomial time. Furthermore, we find the optimal $χ$-bounding function for this larger class of graphs, and we also verify Hadwiger's conjecture for it.

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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

The (theta, wheel)-free graphs Part IV: induced paths and cycles

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class.

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arXiv2019

The (theta, wheel)-free graphs Part III: cliques, stable sets and coloring

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins, and consequently obtain a polynomial time recognition algorithm for the class. In this paper we further use this decomposition theorem to obtain polynomial time algorithms for maximum weight clique, maximum weight stable set and coloring problems. We also show that for a graph $G$ in the class, if its maximum clique size is $ω$, then its chromatic number is bounded by max$\{ω,3\}$, and that the class is 3-clique-colorable.

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arXiv2019

The (theta, wheel)-free graphs Part II: structure theorem

A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this paper we obtain a decomposition theorem for the class of graphs that do not contain an induced subgraph isomorphic to a theta or a wheel, i.e. the class of (theta, wheel)-free graphs. The decomposition theorem uses clique cutsets and 2-joins. Clique cutsets are vertex cutsets that work really well in decomposition based algorithms, but are unfortunately not general enough to decompose more complex hereditary graph classes. A 2-join is an edge cutset that appeared in decomposition theorems of several complex classes, such as perfect graphs, even-hole-free graphs and others. In these decomposition theorems 2-joins are used together with vertex cutsets that are more general than clique cutsets, such as star cutsets and their generalizations (which are much harder to use in algorithms). This is a first example of a decomposition theorem that uses just the combination of clique cutsets and 2-joins. This has several consequences. First, we can easily transform our decomposition theorem into a complete structure theorem for (theta, wheel)-free graphs, i.e. we show how every (theta, wheel)-free graph can be built starting from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations; and all graphs built this way are (theta, wheel)-free. Such structure theorems are very rare for hereditary graph classes, only a few examples are known. Secondly, we obtain an $\mathcal O (n^4m)$-time decomposition based recognition algorithm for (theta, wheel)-free graphs. Finally, in Parts III and IV of this series, we give further applications of our decomposition theorem.

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arXiv2017

On rank-width of even-hole-free graphs

We present a class of (diamond, even hole)-free graphs with no clique cutset that has unbounded rank-width. In general, even-hole-free graphs have unbounded rank-width, because chordal graphs are even-hole-free. A.A. da Silva, A. Silva and C. Linhares-Sales (2010) showed that planar even-hole-free graphs have bounded rank-width, and N.K. Le (2016) showed that even-hole-free graphs with no star cutset have bounded rank-width. A natural question is to ask, whether even-hole-free graphs with no clique cutsets have bounded rank-width. Our result gives a negative answer. Hence we cannot apply Courcelle and Makowsky's meta-theorem which would provide efficient algorithms for a large number of problems, including the maximum independent set problem, whose complexity remains open for (diamond, even hole)-free graphs.

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arXiv2017

The (theta, wheel)-free graphs Part I: only-prism and only-pyramid graphs

Truemper configurations are four types of graphs (namely thetas, wheels, prisms and pyramids) that play an important role in the proof of several decomposition theorems for hereditary graph classes. In this paper, we prove two structure theorems: one for graphs with no thetas, wheels and prisms as induced subgraphs, and one for graphs with no thetas, wheels and pyramids as induced subgraphs. A consequence is a polynomial time recognition algorithms for these two classes. In Part II of this series we generalize these results to graphs with no thetas and wheels as induced subgraphs, and in Parts III and IV, using the obtained structure, we solve several optimization problems for these graphs.

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arXiv2016

Structure and algorithms for (cap, even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $\frac{3}{2}ω(G) -1$, and hence $χ(G)\leq \frac{3}{2}ω(G)$, where $ω(G)$ denotes the size of a largest clique in $G$ and $χ(G)$ denotes the chromatic number of $G$. We give an $O(nm)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O(nm)$ algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6ω(G)-1$ and clique-width at most 48.

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