arXiv subjects

Daniël Paulusma

Publications and source records attributed to Daniël Paulusma.

78 records · Page 5Linked to original sources

arXiv2014

Narrowing the Complexity Gap for Colouring ($C_s$,$P_t$)-Free Graphs

For a positive integer $k$ and graph $G=(V,E)$, a $k$-colouring of $G$ is a mapping $c: V\rightarrow\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The $k$-Colouring problem is to decide, for a given $G$, whether a $k$-colouring of $G$ exists. The $k$-Precolouring Extension problem is to decide, for a given $G=(V,E)$, whether a colouring of a subset of $V$ can be extended to a $k$-colouring of $G$. A $k$-list assignment of a graph is an allocation of a list -a subset of $\{1,\ldots,k\}$- to each vertex, and the List $k$-Colouring problem is to decide, for a given $G$, whether $G$ has a $k$-colouring in which each vertex is coloured with a colour from its list. We continued the study of the computational complexity of these three decision problems when restricted to graphs that contain neither a cycle on $s$ vertices nor a path on $t$ vertices as induced subgraphs (for fixed positive integers $s$ and~$t$).

cs.CC↗

arXiv2014

Induced Disjoint Paths in Circular-Arc Graphs in Linear Time

The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (s_i,t_i) contains paths P_1,...,P_k such that P_i connects s_i and t_i for i=1,...,k, and P_i and P_j have neither common vertices nor adjacent vertices (except perhaps their ends) for 1<=i < j<=k. We present a linear-time algorithm for Induced Disjoint Paths on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs (s_i,t_i) are not necessarily distinct.

cs.DS↗

arXiv2014

Classifying the Clique-Width of $H$-Free Bipartite Graphs

Let $G$ be a bipartite graph, and let $H$ be a bipartite graph with a fixed bipartition $(B_H,W_H)$. We consider three different, natural ways of forbidding $H$ as an induced subgraph in $G$. First, $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Second, $G$ is strongly $H$-free if $G$ is $H$-free or else has no bipartition $(B_G,W_G)$ with $B_H\subseteq B_G$ and $W_H\subseteq W_G$. Third, $G$ is weakly $H$-free if $G$ is $H$-free or else has at least one bipartition $(B_G,W_G)$ with $B_H\not\subseteq B_G$ or $W_H\not\subseteq W_G$. Lozin and Volz characterized all bipartite graphs $H$ for which the class of strongly $H$-free bipartite graphs has bounded clique-width. We extend their result by giving complete classifications for the other two variants of $H$-freeness.

cs.DM↗

arXiv2013

Parameterized Algorithms for Finding Square Roots

We show that the following two problems are fixed-parameter tractable with parameter k: testing whether a connected n-vertex graph with m edges has a square root with at most n-1+k edges and testing whether such a graph has a square root with at least m-k edges. Our first result implies that squares of graphs obtained from trees by adding at most k edges can be recognized in polynomial time for every fixed k>=0; previously this result was known only for k=0. Our second result is equivalent to stating that deciding whether a graph can be modified into a square root of itself by at most k edge deletions is fixed-parameter tractable with parameter k.

cs.DS↗

arXiv2013

Linear-Time Algorithms for Scattering Number and Hamilton-Connectivity of Interval Graphs

Hung and Chang showed that for all k>=1 an interval graph has a path cover of size at most k if and only if its scattering number is at most k. They also showed that an interval graph has a Hamilton cycle if and only if its scattering number is at most 0. We complete this characterization by proving that for all k<=-1 an interval graph is -(k+1)-Hamilton-connected if and only if its scattering number is at most k. We also give an O(m+n) time algorithm for computing the scattering number of an interval graph with n vertices an m edges, which improves the O(n^4) time bound of Kratsch, Kloks and Müller. As a consequence of our two results the maximum k for which an interval graph is k-Hamilton-connected can be computed in O(m+n) time.

cs.DS↗

arXiv2012

Finding vertex-surjective graph homomorphisms

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H, respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k.

cs.DM↗