arXiv subjects

Van Bang Le

Publications and source records attributed to Van Bang Le.

At least 19 recordsLinked to original sources

arXiv2025

The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability

This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. [COCOON 2024]. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection $k$-colorable if it admits a (proper) vertex $k$-coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph $G$ and an integer $k$, whether $G$ is strongly conflict-free $k$-colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption NP $\not\subseteq$ coNP/poly, deciding, for a given graph $G$, whether $G$ is strongly conflict-free $3$-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal $k$-Coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.

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arXiv2025

Complexity and algorithms for matching cut problems in graphs without long induced paths and cycles

In a graph, a (perfect) matching cut is an edge cut that is a (perfect) matching. Matching Cut (MC), respectively, Perfect Matching Cut (PMC), is the problem of deciding whether a given graph has a matching cut, respectively, a perfect matching cut. The Disconnected Perfect Matching problem (DPM) is to decide if a graph has a perfect matching that contains a matching cut. Solving an open problem posed in [Lucke, Paulusma, Ries (ISAAC 2022, Algorithmica 2023)], we show that PMC is NP-complete in graphs without induced 14-vertex path $P_{14}$. Our reduction also works simultaneously for MC and DPM, improving the previous hardness results of MC on $P_{15}$-free graphs and of DPM on $P_{19}$-free graphs to $P_{14}$-free graphs for both problems. Actually, we prove a slightly stronger result: within $P_{14}$-free 8-chordal graphs (graphs without chordless cycles of length at least 9), it is hard to distinguish between those without matching cuts (respectively, perfect matching cuts, disconnected perfect matchings) and those in which every matching cut is a perfect matching cut. Moreover, assuming the Exponential Time Hypothesis, none of these problems can be solved in $2^{o(n)}$ time for $n$-vertex $P_{14}$-free 8-chordal graphs. On the positive side, we show that, as for MC [Moshi (JGT 1989)], DPM and PMC are polynomially solvable when restricted to 4-chordal graphs. Together with the negative results, this partly answers an open question on the complexity of PMC in $k$-chordal graphs asked in [Le, Telle (WG 2021, TCS 2022) & Lucke, Paulusma, Ries (MFCS 2023, TCS 2024)].

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arXiv2024

The complexity of strong conflict-free vertex-connection $k$-colorability

We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be strongly conflict-free vertex-connection $k$-colorable if $G$ admits a vertex $k$-coloring such that any two distinct vertices of $G$ are connected by a conflict-free $shortest$ path. Among others, we show that deciding whether a given graph is strongly conflict-free vertex-connection $3$-colorable is NP-complete even when restricted to $3$-colorable graphs with diameter $3$, radius $2$ and domination number $3$, and, assuming the Exponential Time Hypothesis (ETH), cannot be solved in $2^{o(n)}$ time on such restricted input graphs with $n$ vertices. This hardness result is quite strong when compared to the ordinary $3$-COLORING problem: it is known that $3$-COLORING is solvable in polynomial time in graphs with bounded domination number, and assuming ETH, cannot be solved in $2^{o(\sqrt{n})}$ time in $n$-vertex graphs with diameter $3$ and radius $2$. On the positive side, we point out that a strong conflict-free vertex-connection coloring with minimum color number of a given split graph or a co-bipartite graph can be computed in polynomial time.

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arXiv2024

On polynomial kernelization for Stable Cutset

A stable cutset in a graph $G$ is a set $S\subseteq V(G)$ such that vertices of $S$ are pairwise non-adjacent and such that $G-S$ is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is $NP$-complete to determine whether a given graph $G$ has any stable cutset. Recently, Rauch et al.\ [FCT 2023] gave a number of fixed-parameter tractable (FPT) algorithms, time $f(k)\cdot |V(G)|^c$, for Stable Cutset under a variety of parameters $k$ such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to $P_5$-free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size. We complement these findings by giving the first results on the existence of polynomial kernelizations for \stablecutset, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that $NP\nsubseteq coNP/poly$, we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given $(G,k)$ answering whether there is a stable cutset of size at most $k$. On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.

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arXiv2024

Complexity of the (Connected) Cluster Vertex Deletion problem on $H$-free graphs

The well-known Cluster Vertex Deletion problem (CVD) asks for a given graph $G$ and an integer $k$ whether it is possible to delete a set $S$ of at most $k$ vertices of $G$ such that the resulting graph $G-S$ is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs $H$ for which CVD on $H$-free graphs is polynomially solvable and for which it is NP-complete. Moreover, in the NP-completeness cases, CVD cannot be solved in sub-exponential time in the vertex number of the $H$-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of CVD, the Connected Cluster Vertex Deletion problem (CCVD), in which the set $S$ has to induce a connected subgraph of $G$. It turns out that CCVD admits the same complexity dichotomy for $H$-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on $H$-free graphs.

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arXiv2023

Maximizing Matching Cuts

A matching cut in a graph G is an edge cut of G that is also a matching. This short survey gives an overview of old and new results and open problems for Maximum Matching Cut, which is to determine the size of a largest matching cut in a graph. We also compare this problem with the related problems Matching Cut, Minimum Matching Cut, and Perfect Matching Cut, which are to determine if a graph has a matching cut; the size of a smallest matching cut in a graph; and if a graph has a matching cut that is a perfect matching, respectively. Moreover, we discuss a relationship between Maximum Matching Cut and Max Cut, which is to determine the size of a largest edge cut in a graph, as well as a relationship between Minimum Matching Cut and Min Cut, which is to determine the size of a smallest edge cut in a graph.

math.CO↗

arXiv2023

Computing Optimal Leaf Roots of Chordal Cographs in Linear Time

A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G if and only if the distance between x and y in T is at most k. Such a tree T is called a k-leaf root of G. The computational problem of constructing a k-leaf root for a given graph G and an integer k, if any, is motivated by the challenge from computational biology to reconstruct phylogenetic trees. For fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time. In this paper, we propose to study optimal leaf roots of graphs G, that is, the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots is more justified as they yield more probable phylogenetic trees. Lafond's result does not imply polynomial-time computability of optimal leaf roots, because, even for optimal k-leaf roots, k may (exponentially) depend on the size of G. This paper presents a linear-time construction of optimal leaf roots for chordal cographs (also known as trivially perfect graphs). Additionally, it highlights the importance of the parity of the parameter k and provides a deeper insight into the differences between optimal k-leaf roots of even versus odd k. Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect leaf power, chordal cograph

math.CO↗

arXiv2022

On the $d$-Claw Vertex Deletion Problem

Let $d$-claw (or $d$-star) stand for $K_{1,d}$, the complete bipartite graph with 1 and $d\ge 1$ vertices on each part. The $d$-claw vertex deletion problem, $d$-CLAW-VD, asks for a given graph $G$ and an integer $k$ if one can delete at most $k$ vertices from $G$ such that the resulting graph has no $d$-claw as an induced subgraph. Thus, 1-CLAW-VD and 2-CLAW-VD are just the famous VERTEX COVER problem and the CLUSTER VERTEX DELETION problem, respectively. In this paper, we strengthen a hardness result in [M. Yannakakis, Node-Deletion Problems on Bipartite Graphs, SIAM J. Comput. (1981)], by showing that CLUSTER VERTEX DELETION remains NP-complete when restricted to bipartite graphs of maximum degree 3. Moreover, for every $d\ge 3$, we show that $d$-CLAW-VD is NP-complete even when restricted to bipartite graphs of maximum degree $d$. These hardness results are optimal with respect to degree constraint. By extending the hardness result in [F. Bonomo-Braberman et al., Linear-Time Algorithms for Eliminating Claws in Graphs, COCOON 2020], we show that, for every $d\ge 3$, $d$-CLAW-VD is NP-complete even when restricted to split graphs without $(d+1)$-claws, and split graphs of diameter 2. On the positive side, we prove that $d$-CLAW-VD is polynomially solvable on what we call $d$-block graphs, a class properly contains all block graphs. This result extends the polynomial-time algorithm in [Y. Cao et al., Vertex deletion problems on chordal graphs, Theor. Comput. Sci. (2018)] for 2-CLAW-VD on block graphs to $d$-CLAW-VD for all $d\ge 2$ and improves the polynomial-time algorithm proposed by F. Bonomo-Brabeman et al. for (unweighted) 3-CLAW-VD on block graphs to 3-block graphs.

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arXiv2021

The Perfect Matching Cut Problem Revisited

In a graph, a perfect matching cut is an edge cut that is a perfect matching. Perfect Matching Cut (PMC) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP-complete. We revisit the problem and show that PMC remains NP-complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which PMC is polynomial time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no $O^*(2^{o(n)})$-time algorithm for PMC even when restricted to $n$-vertex bipartite graphs, and also show that PMC can be solved in $O^*(1.2721^n)$ time by means of an exact branching algorithm.

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arXiv2021

Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration

An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

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arXiv2019

Color-line and Proper Color-line Graphs

Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion of color-line graphs that generalizes the classical concept of line graphs in a natural way. Let $H$ be a (properly) edge-colored graph. The (proper) color-line graph $C\!L(H)$ of $H$ has edges of $H$ as vertices, and two edges of $H$ are adjacent in $C\!L(H)$ if they are incident in $H$ or have the same color. We give Krausz-type characterizations for (proper) color-line graphs, and point out that, for any fixed $k\ge 2$, recognizing if a graph is the color-line graph of some graph $H$ in which the edges are colored with at most $k$ colors is NP-complete. In contrast, we show that, for any fixed $k$, recognizing color-line graphs of properly edge colored graphs $H$ with at most $k$ colors is polynomially. Moreover, we give a good characterization for proper $2$-color line graphs that yields a linear time recognition algorithm in this case.

math.CO↗

arXiv2018

Map graphs having witnesses of large girth

A half-square of a bipartite graph $B=(X,Y,E_B)$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. If $G=(V,E_G)$ is the half-square of a planar bipartite graph $B=(V,W,E_B)$, then $G$ is called a map graph, and $B$ is a witness of $G$. Map graphs generalize planar graphs, and have been introduced and investigated by Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]. They proved that recognizing map graphs is in $\mathsf{NP}$ by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizing map graphs is in $\mathsf{P}$, by providing an $Ω(n^{120})$-time algorithm for $n$-vertex input graphs. In this note, we give good characterizations and efficient recognition for half-squares of bipartite graphs with girth at least a given integer $g\ge 8$. It turns out that map graphs having witnesses of girth at least $g$ are precisely the graphs whose vertex-clique incidence bipartite graph is planar and of girth at least $g$. Our structural characterization implies an $O(n^2m)$-time algorithm for recognizing if a given $n$-vertex $m$-edge graph $G$ is such a map graph.

cs.DM↗

arXiv2018

A complexity dichotomy for Matching Cut in (bipartite) graphs of fixed diameter

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut is NP-complete in the class of graphs of diameter $d$. This resolves an open problem posed by Borowiecki and Jesse-Józefczyk in [Matching cutsets in graphs of diameter $2$, Theoretical Computer Science 407 (2008) 574-582]. We then show that, for any fixed integer $d\ge 4$, Matching Cut is NP-complete even when restricted to the class of bipartite graphs of diameter $d$. Complementing the hardness results, we show that Matching Cut is polynomial-time solvable in the class of bipartite graphs of diameter at most three, and point out a new and simple polynomial-time algorithm solving Matching Cut in graphs of diameter $2$.

cs.CC↗

arXiv2018

Hardness and structural results for half-squares of restricted tree convex bipartite graphs

Let $B=(X,Y,E)$ be a bipartite graph. A half-square of $B$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. Every planar graph is a half-square of a planar bipartite graph, namely of its subdivision. Until recently, only half-squares of planar bipartite graphs, also known as map graphs (Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]), have been investigated, and the most discussed problem is whether it is possible to recognize these graphs faster and simpler than Thorup's $O(n^{120})$-time algorithm (Thorup [FOCS 1998]). In this paper, we identify the first hardness case, namely that deciding if a graph is a half-square of a balanced bisplit graph is NP-complete. (Balanced bisplit graphs form a proper subclass of star convex bipartite graphs.) For classical subclasses of tree convex bipartite graphs such as biconvex, convex, and chordal bipartite graphs, we give good structural characterizations of their half-squares that imply efficient recognition algorithms. As a by-product, we obtain new characterizations of unit interval graphs, interval graphs, and of strongly chordal graphs in terms of half-squares of biconvex bipartite, convex bipartite, and of chordal bipartite graphs, respectively. Good characterizations of half-squares of star convex and star biconvex bipartite graphs are also given, giving linear-time recognition algorithms for these half-squares.

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arXiv2016

On the complete width and edge clique cover problems

A complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the graph $G'$ obtained from $G$ by adding some new edges between certain vertices inside the sets $\mathtt{N}_i$, $1\le i\le k$, is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most $k$ or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete on $3K_2$-free bipartite graphs and polynomially solvable on $2K_2$-free bipartite graphs and on $(2K_2,C_4)$-free graphs. As a by-product, we obtain the following new results: the edge clique cover problem is NP-complete on $\overline{3K_2}$-free co-bipartite graphs and polynomially solvable on $C_4$-free co-bipartite graphs and on $(2K_2, C_4)$-free graphs. We also give a characterization for $k$-probe complete graphs which implies that the complete width problem admits a kernel of at most $2^k$ vertices. This provides another proof for the known fact that the edge clique cover problem admits a kernel of at most $2^k$ vertices. Finally we determine all graphs of small complete width $k\le 3$.

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arXiv2016

Good characterizations and linear time recognition for 2-probe block graphs

Block graphs are graphs in which every block (biconnected component) is a clique. A graph $G=(V,E)$ is said to be an (unpartitioned) $k$-probe block graph if there exist $k$ independent sets $N_i\subseteq V$, $1\le i\le k$, such that the graph $G'$ obtained from $G$ by adding certain edges between vertices inside the sets $N_i$, $1\le i\le k$, is a block graph; if the independent sets $N_i$ are given, $G$ is called a partitioned $k$-probe block graph. In this paper we give good characterizations for $2$-probe block graphs, in both unpartitioned and partitioned cases. As an algorithmic implication, partitioned and unpartitioned probe block graphs can be recognized in linear time, improving a recognition algorithm of cubic time complexity previously obtained by Chang et al. [Block-graph width, Theoretical Computer Science 412 (2011), 2496--2502].

math.CO↗

arXiv2015

Characterization and recognition of some opposition and coalition graph classes

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs are long-standing open problems. This paper gives characterizations for opposition graphs and coalition graphs on some restricted graph classes. Implicit in our arguments are polynomial time recognition algorithms for these graphs. We also give a good characterization for the so-called generalized opposition graphs.

cs.DM↗

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.

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