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Oliver Schaudt

Publications and source records attributed to Oliver Schaudt.

31 records · Page 2Linked to original sources
arXiv2015

Recognizing k-equistable graphs in FPT time

A graph $G = (V,E)$ is called equistable if there exist a positive integer $t$ and a weight function $w : V \to \mathbb{N}$ such that $S \subseteq V$ is a maximal stable set of $G$ if and only if $w(S) = t$. Such a function $w$ is called an equistable function of $G$. For a positive integer $k$, a graph $G = (V,E)$ is said to be $k$-equistable if it admits an equistable function which is bounded by $k$. We prove that the problem of recognizing $k$-equistable graphs is fixed parameter tractable when parameterized by $k$, affirmatively answering a question of Levit et al. In fact, the problem admits an $O(k^5)$-vertex kernel that can be computed in linear time.

cs.DS↗
arXiv2014

Squares of $3$-sun-free split graphs

The square of a graph $G$, denoted by $G^2$, is obtained from $G$ by putting an edge between two distinct vertices whenever their distance is two. Then $G$ is called a square root of $G^2$. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a split graph, that is, a graph in which the vertex set can be partitioned into a stable set and a clique. We give a wide range of polynomial time solvable cases for the problem of recognizing if a given graph is the square of some special kind of split graph. To the best of our knowledge, our result properly contains all previously known such cases. Our polynomial time algorithms are build on a structural investigation of graphs that admit a split square root that is 3-sun-free, and may pave the way toward a dichotomy theorem for recognizing squares of (3-sun-free) split graphs.

cs.CC↗
arXiv2014

Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded. From this we derive that the whole graph, if large enough, may be covered with 14 vertex-disjoint monochromatic cycles.

math.CO↗
arXiv2014

A new characterization of $P_k$-free graphs

The class of graphs that do not contain an induced path on $k$ vertices, $P_k$-free graphs, plays a prominent role in algorithmic graph theory. This motivates the search for special structural properties of $P_k$-free graphs, including alternative characterizations. Let $G$ be a connected $P_k$-free graph, $k \ge 4$. We show that $G$ admits a connected dominating set whose induced subgraph is either $P_{k-2}$-free, or isomorphic to $P_{k-2}$. Surprisingly, it turns out that every minimum connected dominating set of $G$ has this property. This yields a new characterization for $P_k$-free graphs: a graph $G$ is $P_k$-free if and only if each connected induced subgraph of $G$ has a connected dominating set whose induced subgraph is either $P_{k-2}$-free, or isomorphic to $C_k$. This improves and generalizes several previous results; the particular case of $k=7$ solves a problem posed by van 't Hof and Paulusma [A new characterization of $P_6$-free graphs, COCOON 2008]. In the second part of the paper, we present an efficient algorithm that, given a connected graph $G$ on $n$ vertices and $m$ edges, computes a connected dominating set $X$ of $G$ with the following property: for the minimum $k$ such that $G$ is $P_k$-free, the subgraph induced by $X$ is $P_{k-2}$-free or isomorphic to $P_{k-2}$. As an application our results, we prove that Hypergraph 2-Colorability, an NP-complete problem in general, can be solved in polynomial time for hypergraphs whose vertex-hyperedge incidence graph is $P_7$-free.

cs.DM↗
arXiv2014

Structural parameterizations for boxicity

The boxicity of a graph $G$ is the least integer $d$ such that $G$ has an intersection model of axis-aligned $d$-dimensional boxes. Boxicity, the problem of deciding whether a given graph $G$ has boxicity at most $d$, is NP-complete for every fixed $d \ge 2$. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive $1$-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.

math.CO↗
arXiv2014

Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs

The square of a graph $G$, denoted $G^2$, is obtained from $G$ by putting an edge between two distinct vertices whenever their distance is two. Then $G$ is called a square root of $G^2$. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a chordal graph or even a split graph. We present a polynomial time algorithm that decides whether a given graph has a ptolemaic square root. If such a root exists, our algorithm computes one with a minimum number of edges. In the second part of our paper, we give a characterization of the graphs that admit a 3-sun-free split square root. This characterization yields a polynomial time algorithm to decide whether a given graph has such a root, and if so, to compute one.

cs.DM↗
arXiv2014

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by χ_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = χ(G),\ldots,χ_b(G), and b-monotonic if χ_b(H_1) \geq χ_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.

cs.CC↗
arXiv2013

The graph formulation of the union-closed sets conjecture

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph there are two adjacent vertices each belonging to at most half of the maximal stable sets. In this graph formulation other special cases become natural. The conjecture is trivially true for non-bipartite graphs and we show that it holds also for the classes of chordal bipartite graphs, subcubic bipartite graphs, bipartite series-parallel graphs and bipartitioned circular interval graphs.

math.CO↗
arXiv2013

The Price of Connectivity for Vertex Cover

The vertex cover number of a graph is the minimum number of vertices that are needed to cover all edges. When those vertices are further required to induce a connected subgraph, the corresponding number is called the connected vertex cover number, and is always greater or equal to the vertex cover number. Connected vertex covers are found in many applications, and the relationship between those two graph invariants is therefore a natural question to investigate. For that purpose, we introduce the {\em Price of Connectivity}, defined as the ratio between the two vertex cover numbers. We prove that the price of connectivity is at most 2 for arbitrary graphs. We further consider graph classes in which the price of connectivity of every induced subgraph is bounded by some real number $t$. We obtain forbidden induced subgraph characterizations for every real value $t \leq 3/2$. We also investigate critical graphs for this property, namely, graphs whose price of connectivity is strictly greater than that of any proper induced subgraph. Those are the only graphs that can appear in a forbidden subgraph characterization for the hereditary property of having a price of connectivity at most $t$. In particular, we completely characterize the critical graphs that are also chordal. Finally, we also consider the question of computing the price of connectivity of a given graph. Unsurprisingly, the decision version of this question is NP-hard. In fact, we show that it is even complete for the class $Θ_2^P = P^{NP[\log]}$, the class of decision problems that can be solved in polynomial time, provided we can make $O(\log n)$ queries to an NP-oracle. This paves the way for a thorough investigation of the complexity of problems involving ratios of graph invariants.

cs.DM↗
arXiv2013

A Note on Connected Dominating Set in Graphs Without Long Paths And Cycles

The ratio of the connected domination number, $γ_c$, and the domination number, $γ$, is strictly bounded from above by 3. It was shown by Zverovich that for every connected $(P_5,C_5)$-free graph, $γ_c = γ$. In this paper, we investigate the interdependence of $γ$ and $γ_c$ in the class of $(P_k,C_k)$-free graphs, for $k \ge 6$. We prove that for every connected $(P_6,C_6)$-free graph, $γ_c \le γ+ 1$ holds, and there is a family of $(P_6,C_6)$-free graphs with arbitrarily large values of $γ$ attaining this bound. Moreover, for every connected $(P_8,C_8)$-free graph, $γ_c / γ\le 2$, and there is a family of $(P_7,C_7)$-free graphs with arbitrarily large values of $γ$ attaining this bound. In the class of $(P_9,C_9)$-free graphs, the general bound $γ_c / γ\le 3$ is asymptotically sharp.

cs.DM↗
arXiv2013

The union-closed sets conjecture almost holds for almost all random bipartite graphs

Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an equivalent formulation in terms of graphs: In every bipartite graph with least one edge, both colour classes contain a vertex belonging to at most half of the maximal stable sets. We prove that, for every fixed edge-probability, almost every random bipartite graph almost satisfies Frankl's conjecture.

math.CO↗