arXiv subjects

Daniel Paulusma

Publications and source records attributed to Daniel Paulusma.

At least 37 records · Page 2Linked to original sources

arXiv2020

Colouring $(sP_1+P_5)$-Free Graphs: a Mim-Width Perspective

We prove that the class of $(K_t,sP_1+P_5)$-free graphs has bounded mim-width for every $s\geq 0$ and $t\geq 1$, and that there is a polynomial-time algorithm that, given a graph in the class, computes a branch decomposition of constant mim-width. A large number of \NP-complete graph problems become polynomial-time solvable on graph classes with bounded mim-width and for which a branch decomposition is quickly computable. The $k$-Colouring problem is an example of such a problem. For this problem, we may assume that the input graph is $K_{k+1}$-free. Then, as a consequence of our result, we obtain a new proof for the known result that for every fixed $k\geq 1$ and $s\geq 0$, $k$-Colouring is polynomial-time solvable for $(sP_1+P_5)$-free graphs. In fact, our findings show that the underlying reason for this polynomial-time algorithm is that the class has bounded mim-width.

cs.DS↗

arXiv2019

Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs

A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a subset of $\{1, 2, \dots\}$ of size $k$. A colouring $c$ of $G$ respects a $k$-regular list assignment $L$ of $G$ if $c(u)\in L(u)$ for every $u\in V$. A graph $G$ is $k$-choosable if for every $k$-regular list assignment $L$ of $G$, there exists a colouring of $G$ that respects $L$. We may also ask if for a given $k$-regular list assignment $L$ of a given graph $G$, there exists a colouring of $G$ that respects $L$. This yields the $k$-Regular List Colouring problem. For $k\in \{3,4\}$ we determine a family of classes ${\cal G}$ of planar graphs, such that either $k$-Regular List Colouring is NP-complete for instances $(G,L)$ with $G\in {\cal G}$, or every $G\in {\cal G}$ is $k$-choosable. By using known examples of non-$3$-choosable and non-$4$-choosable graphs, this enables us to classify the complexity of $k$-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no $4$-cycles and no $5$-cycles. We also classify the complexity of $k$-Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.

cs.DS↗

arXiv2018

Classifying k-Edge Colouring for H-free Graphs

A graph is $H$-free if it does not contain an induced subgraph isomorphic to $H$. For every integer $k$ and every graph $H$, we determine the computational complexity of $k$-Edge Colouring for $H$-free graphs.

cs.DS↗

arXiv2018

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

cs.DS↗

arXiv2018

Contracting to a Longest Path in H-Free Graphs

We prove two dichotomy results for detecting long paths as patterns in a given graph. The NP-hard problem Longest Induced Path is to determine the longest induced path in a graph. The NP-hard problem Longest Path Contractibility is to determine the longest path to which a graph can be contracted to. By combining known results with new results we completely classify the computational complexity of both problems for $H$-free graphs. Our main focus is on the second problem, for which we design a general contractibility technique that enables us to reduce the problem to a matching problem.

cs.DS↗

arXiv2018

Finding a Small Number of Colourful Components

A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL PARTITION problem is to decide whether a coloured graph $(G,c)$ has a colourful partition of size at most $k$. This problem is closely related to the COLOURFUL COMPONENTS problem, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most $p$ edges. Nevertheless we show that COLOURFUL PARTITION and COLOURFUL COMPONENTS may have different complexities for restricted instances. We tighten known NP-hardness results for both problems and in addition we prove new hardness and tractability results for COLOURFUL PARTITION. Using these results we complete our paper with a thorough parameterized study of COLOURFUL PARTITION.

cs.DS↗

arXiv2018

Using Contracted Solution Graphs for Solving Reconfiguration Problems

We introduce in a general setting a dynamic programming method for solving reconfiguration problems. Our method is based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. Our general framework captures the approach behind known reconfiguration results of Bonsma (2012) and Hatanaka, Ito and Zhou (2014). As a third example, we apply the method to the following problem: given two $k$-colorings $α$ and $β$ of a graph $G$, can $α$ be modified into $β$ by recoloring one vertex of $G$ at a time, while maintaining a $k$-coloring throughout? This problem is known to be PSPACE-hard even for bipartite planar graphs and $k=4$. By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial time algorithm for $(k-2)$-connected chordal graphs.

cs.DS↗

arXiv2018

Connected Vertex Cover for $(sP_1+P_5)$-Free Graphs

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most $k$ that induces a connected subgraph of $G$. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for $H$-free graphs if $H$ is not a linear forest (a graph is $H$-free if it does not contain $H$ as an induced subgraph). It is easy to see that Connected Vertex Cover is polynomial-time solvable for $P_4$-free graphs. We continue the search for tractable graph classes: we prove that it is also polynomial-time solvable for $(sP_1+P_5)$-free graphs for every integer $s\geq 0$.

cs.DS↗

arXiv2018

Disconnected Cuts in Claw-free Graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The decision problem whether a graph has a disconnected cut is called Disconnected Cut. This problem is closely related to several homomorphism and contraction problems, and fits in an extensive line of research on vertex cuts with additional properties. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms are known for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle $C_4$ led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The centerpiece of our result is a novel decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and expands the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further novel algorithmic results, namely Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

cs.DS↗

arXiv2017

Surjective H-Colouring over Reflexive Digraphs

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete. By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

cs.CC↗

arXiv2017

On Colouring $(2P_2,H)$-Free and $(P_5,H)$-Free Graphs

The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no induced subgraph isomorphic to $H_1$ or $H_2$. A connected graph $H_1$ is almost classified if Colouring on $(H_1,H_2)$-free graphs is known to be polynomial-time solvable or NP-complete for all but finitely many connected graphs $H_2$. We show that every connected graph $H_1$ apart from the claw $K_{1,3}$ and the $5$-vertex path $P_5$ is almost classified. We also prove a number of new hardness results for Colouring on $(2P_2,H)$-free graphs. This enables us to list all graphs $H$ for which the complexity of Colouring is open on $(2P_2,H)$-free graphs and all graphs $H$ for which the complexity of Colouring is open on $(P_5,H)$-free graphs. In fact we show that these two lists coincide. Moreover, we show that the complexities of Colouring for $(2P_2,H)$-free graphs and for $(P_5,H)$-free graphs are the same for all known cases.

cs.CC↗

arXiv2017

Independent Feedback Vertex Sets for Graphs of Bounded Diameter

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets $A$ and $B$, where $A$ is an independent set and $B$ induces a forest. The set $A$ in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

cs.DS↗

arXiv2017

Independent Feedback Vertex Set for $P_5$-free Graphs

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $k\geq 0$, to delete at most $k$ vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the $3$-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for $H$-free graphs. We prove that it is NP-complete if $H$ contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $P_4$-free graphs. We show that it remains polynomial-time solvable for $P_5$-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most $k$ for a given integer $k\geq 0$. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for $H$-free graphs with the complexity of $3$-Colouring, Independent Odd Cycle Transversal and other related problems.

cs.DS↗

arXiv2017

Surjective H-Colouring: New Hardness Results

A homomorphism from a graph G to a graph H is a vertex mapping f from the vertex set of G to the vertex set of H such that there is an edge between vertices f(u) and f(v) of H whenever there is an edge between vertices u and v of G. The H-Colouring problem is to decide whether or not a graph G allows a homomorphism to a fixed graph H. We continue a study on a variant of this problem, namely the Surjective H-Colouring problem, which imposes the homomorphism to be vertex-surjective. We build upon previous results and show that this problem is NP-complete for every connected graph H that has exactly two vertices with a self-loop as long as these two vertices are not adjacent. As a result, we can classify the computational complexity of Surjective H-Colouring for every graph H on at most four vertices.

cs.CC↗

arXiv2016

New Bounds for the Snake-in-the-Box Problem

The Snake-in-the-Box problem is that of finding a longest induced path in an $n$-dimensional hypercube. We prove new lower bounds for the values $n\in \{11,12,13\}$. The Coil-in-the-Box problem is that of finding a longest induced cycle in an $n$-dimensional hypercube. We prove new lower bounds for the values $n\in \{12,13\}$.

math.CO↗

arXiv2015

Editing to a Planar Graph of Given Degrees

We consider the following graph modification problem. Let the input consist of a graph $G=(V,E)$, a weight function $w\colon V\cup E\rightarrow \mathbb{N}$, a cost function $c\colon V\cup E\rightarrow \mathbb{N}$ and a degree function $δ\colon V\rightarrow \mathbb{N}_0$, together with three integers $k_v, k_e$ and $C$. The question is whether we can delete a set of vertices of total weight at most $k_v$ and a set of edges of total weight at most $k_e$ so that the total cost of the deleted elements is at most $C$ and every non-deleted vertex $v$ has degree $δ(v)$ in the resulting graph $G'$. We also consider the variant in which $G'$ must be connected. Both problems are known to be NP-complete and W[1]-hard when parameterized by $k_v+k_e$. We prove that, when restricted to planar graphs, they stay NP-complete but have polynomial kernels when parameterized by $k_v+k_e$.

cs.DS↗

arXiv2015

Kempe Equivalence of Colourings of Cubic Graphs

Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chain is a subset of $V$ that induces a maximal connected subgraph of $G$ in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for $k \geq 3$, all $k$-colourings of $k$-regular graphs that are not complete are Kempe equivalent. We address the case $k=3$ by showing that all $3$-colourings of a cubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ or the triangular prism.

cs.DM↗

arXiv2014

The Computational Complexity of Disconnected Cut and 2K2-Partition

For a connected graph G=(V,E), a subset U of V is called a disconnected cut if U disconnects the graph and the subgraph induced by U is disconnected as well. We show that the problem to test whether a graph has a disconnected cut is NP-complete. This problem is polynomially equivalent to the following problems: testing if a graph has a 2K2-partition, testing if a graph allows a vertex-surjective homomorphism to the reflexive 4-cycle and testing if a graph has a spanning subgraph that consists of at most two bicliques. Hence, as an immediate consequence, these three decision problems are NP-complete as well. This settles an open problem frequently posed in each of the four settings.

cs.CC↗