arXiv subjects

Andreas Brandstädt

Publications and source records attributed to Andreas Brandstädt.

At least 19 recordsLinked to original sources

arXiv2022

Finding Efficient Domination for $S_{1,3,3}$-Free Bipartite Graphs in Polynomial Time

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is \NP-complete for various $H$-free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least $g$ for every fixed $g$. Thus, ED is \NP-complete for $K_{1,4}$-free bipartite graphs and for $C_4$-free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for $S_{1,3,3}$-free bipartite graphs.

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arXiv2021

Finding Efficient Domination for $(S_{1,2,5},S_{3,3,3}$-Free Chordal Bipartite Graphs in Polynomial Time

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete for chordal bipartite graphs as well as for $P_7$-free graphs, and even for very restricted $H$-free bipartite graph classes such as for $K_{1,4}$-free bipartite graphs as well as for $C_4$-free bipartite graphs while it is solvable in polynomial time for $P_8$-free bipartite graphs as well as for $S_{1,3,3}$-free bipartite graphs and for $S_{1,1,5}$-free bipartite graphs. Here we show that ED can be solved in polynomial time for $(S_{1,2,5},S_{3,3,3})$-free chordal bipartite graphs.

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arXiv2021

Finding Efficient Domination for $P_8$-Free Bipartite Graphs in Polynomial Time

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete for $P_7$-free graphs, and even for very restricted $H$-free bipartite graph classes such as for $K_{1,4}$-free bipartite graphs as well as for $C_4$-free bipartite graphs while it is solvable in polynomial time for $P_7$-free bipartite graphs as well as for $S_{2,2,4}$-free bipartite graphs. Here we show that ED can be solved in polynomial time for $P_8$-free bipartite graphs.

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arXiv2021

Finding Efficient Domination for $S_{1,1,5}$-Free Bipartite Graphs in Polynomial Time

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (e.d.s.\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is \NP-complete for various $H$-free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least $g$ for every fixed $g$. Thus, ED is \NP-complete for $K_{1,4}$-free bipartite graphs and for $C_4$-free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for $S_{1,1,5}$-free bipartite graphs.

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arXiv2020

Finding Dominating Induced Matchings in $P_9$-Free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge subset $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but was solved in linear time for $P_7$-free graphs and in polynomial time for $P_8$-free graphs. In this paper, we solve it in polynomial time for $P_9$-free graphs.

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arXiv2020

Finding Dominating Induced Matchings in $S_{1,1,5}$-Free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for $P_7$-free graphs, and in polynomial time for $S_{1,2,4}$-free graphs as well as for $S_{2,2,2}$-free graphs and for $S_{2,2,3}$-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{1,1,5}$-free graphs.

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arXiv2020

Finding Dominating Induced Matchings in $(S_{2,2,3})$-Free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 and is solvable in linear time for $P_7$-free graphs, and in polynomial time for $S_{1,2,4}$-free graphs as well as for $S_{2,2,2}$-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{2,2,3}$-free graphs.

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arXiv2019

On Efficient Domination for Some Classes of $H$-Free Bipartite Graphs

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete even for very restricted $H$-free graph classes such as for $2P_3$-free chordal graphs while it is solvable in polynomial time for $P_6$-free graphs. Here we focus on $H$-free bipartite graphs: We show that (weighted) ED can be solved in polynomial time for $H$-free bipartite graphs when $H$ is $P_7$ or $\ell P_4$ for fixed $\ell$, and similarly for $P_9$-free bipartite graphs with vertex degree at most 3, and when $H$ is $S_{2,2,4}$. Moreover, we show that ED is \NP-complete for bipartite graphs with diameter at most 6.

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arXiv2019

On Efficient Domination for Some Classes of $H$-Free Chordal Graphs

A vertex set $D$ in a finite undirected graph $G$ is an efficient dominating set (e.d.s. for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete even for very restricted graph classes such as for $2P_3$-free chordal graphs while it is solvable in polynomial time for $P_6$-free chordal graphs (and even for $P_6$-free graphs). A standard reduction from the \NP-complete Exact Cover problem shows that ED is \NP-complete for a very special subclass of chordal graphs generalizing split graphs. The reduction implies that ED is \NP-complete e.g.\ for double-gem-free chordal graphs while it is solvable in linear time for gem-free chordal graphs (by various reasons such as bounded clique-width, distance-hereditary graphs, chordal square etc.), and ED is \NP-complete for butterfly-free chordal graphs while it is solvable in linear time for $2P_2$-free graphs. We show that (weighted) ED can be solved in polynomial time for $H$-free chordal graphs when $H$ is net, extended gem, or $S_{1,2,3}$.

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arXiv2019

Dominating Induced Matchings in $S_{1,2,4}$-Free Graphs

Let $G=(V,E)$ be a finite undirected graph without loops and multiple edges. A subset $M \subseteq E$ of edges is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $M$. In particular, this means that $M$ is an induced matching, and every edge not in $M$ shares exactly one vertex with an edge in $M$. Clearly, not every graph has a d.i.m. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the {\em Efficient Domination} problem for line graphs. The DIM problem is \NP-complete in general, and even for very restricted graph classes such as planar bipartite graphs with maximum degree 3. However, DIM is solvable in polynomial time for claw-free (i.e., $S_{1,1,1}$-free) graphs, for $S_{1,2,3}$-free graphs as well as for $S_{2,2,2}$-free graphs, in linear time for $P_7$-free graphs, and in polynomial time for $P_8$-free graphs ($P_k$ is a special case of $S_{i,j,\ell}$). In a paper by Hertz, Lozin, Ries, Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomial time for $S_{i,j,k}$-free graphs for every fixed $i,j,k$. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{1,2,4}$-free graphs which generalizes the $S_{1,2,3}$-free as well as the $P_7$-free case.

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arXiv2019

Maximum Weight Independent Sets for ($S_{1,2,4}$,Triangle)-Free Graphs in Polynomial Time

The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity for $P_k$-free graphs, $k \ge 7$, is an open problem. In \cite{BraMos2018}, it is shown that MWIS can be solved in polynomial time for ($P_7$,triangle)-free graphs. This result is extended by Maffray and Pastor \cite{MafPas2016} showing that MWIS can be solved in polynomial time for ($P_7$,bull)-free graphs. In the same paper, they also showed that MWIS can be solved in polynomial time for ($S_{1,2,3}$,bull)-free graphs. In this paper, using a similar approach as in \cite{BraMos2018}, we show that MWIS can be solved in polynomial time for ($S_{1,2,4}$,triangle)-free graphs which generalizes the result for ($P_7$,triangle)-free graphs.

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arXiv2017

On Chordal-$k$-Generalized Split Graphs

A graph $G$ is a {\em chordal-$k$-generalized split graph} if $G$ is chordal and there is a clique $Q$ in $G$ such that every connected component in $G[V \setminus Q]$ has at most $k$ vertices. Thus, chordal-$1$-generalized split graphs are exactly the split graphs. We characterize chordal-$k$-generalized split graphs by forbidden induced subgraphs. Moreover, we characterize a very special case of chordal-$2$-generalized split graphs for which the Efficient Domination problem is \NP-complete.

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arXiv2015

Bounding the Clique-Width of $H$-free Chordal Graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex $P_4$-extensions $H$ for which the class of $H$-free chordal graphs has bounded clique-width. In fact we prove that bull-free chordal and co-chair-free chordal graphs have clique-width at most 3 and 4, respectively. In particular, we find four new classes of $H$-free chordal graphs of bounded clique-width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs $H$ for which the class of $H$-free chordal graphs has bounded clique-width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of $(2P_1+P_3,K_4)$-free graphs has bounded clique-width via a reduction to $K_4$-free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique-width of $H$-free weakly chordal graphs.

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arXiv2015

Bounding the Clique-Width of $H$-free Split Graphs

A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We continue a study into the boundedness of clique-width of subclasses of perfect graphs. We identify five new classes of $H$-free split graphs whose clique-width is bounded. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs $H$ for which the class of $H$-free split graphs has bounded clique-width.

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arXiv2015

Efficient Domination for Some Subclasses of $P_6$-Free Graphs in Polynomial Time

Let $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\ in $G$, is known to be \NP-complete even for very restricted graph classes such as $P_7$-free chordal graphs. The ED problem on a graph $G$ can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of $G$. The complexity of the ED problem is an open question for $P_6$-free graphs and was open even for the subclass of $P_6$-free chordal graphs. In this paper, we show that squares of $P_6$-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of ($P_6$, house, hole, domino)-free graphs. This implies that ED/WeightedED is solvable in polynomial time for ($P_6$, house, hole, domino)-free graphs; in particular, for $P_6$-free chordal graphs. Moreover, based on our result that squares of $P_6$-free graphs that have an e.d. are hole-free and some properties concerning odd antiholes, we show that squares of ($P_6$, house)-free graphs (($P_6$, bull)-free graphs, respectively) that have an e.d. are perfect. This implies that ED/WeightedED is solvable in polynomial time for ($P_6$, house)-free graphs and for ($P_6$, bull)-free graphs (the time bound for ($P_6$, house, hole, domino)-free graphs is better than that for ($P_6$, house)-free graphs). The complexity of the ED problem for $P_6$-free graphs remains an open question.

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arXiv2014

Polynomial-time Algorithms for Weighted Efficient Domination Problems in AT-free Graphs and Dually Chordal Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs.

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arXiv2013

The Dilworth Number of Auto-Chordal-Bipartite Graphs

The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E) has the same color classes X and Y as B, and two vertices x in X and y in Y are adjacent in mir(B) if and only if xy is not in E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal bipartite as well; we call these graphs auto-chordal bipartite graphs (ACB graphs for short). We describe the relationship to some known graph classes such as interval and strongly chordal graphs and we present several characterizations of ACB graphs. We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k.

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arXiv2013

New Polynomial Cases of the Weighted Efficient Domination Problem

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3.

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