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 $k\le 3$.
In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.
The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smal…
We present a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. This result provides a negative answer to the question asked by Daligault, Rao and Thomassé in ("Well-quasi-order of relabel functions", Order, 27(3):301--315, 2010).
If a graph has no induced subgraph isomorphic to $H_1$ or $H_2$ then it is said to be ($H_1,H_2$)-free. Dabrowski and Paulusma found 13 open cases for the question whether the clique-width of ($H_1,H_2$)-free graphs is bounded. One of them is the class of ($S_{1,2,2}$,triangle)-free graphs. In this paper we show that these graphs have bounded clique-width. Thus, also ($P_1+2P_2$,triangle)-free graphs have bounded clique-width which solves another open problem of Dabrowski and Paulusma. Meanwhile we were informed by Paulusma that in December 2015, Dabrowski, Dross and Paulusma showed that ($S_{…
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.
Warsaw University of Technology（华沙理工大学）；University of Warsaw（华沙大学）；FAMNIT and IAM, University of Primorska（普里莫斯卡大学 FAMNIT 与 IAM）；IT University of Copenhagen（哥本哈根信息技术大学）
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 bounde…
Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the $|{\cal H}|=1$ case by classifying the boundedness of clique-width for every set ${\cal H}$ of self-complementary graphs. We then completely settle the $|{\cal H}|=2$ case. In particular, we determ…
Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no subgraph isomorphic to $H_1$ or $H_2$. We continue a recent study into the clique-width of $(H_1,H_2)$-free graphs and present three new classes of $(H_1,H_2)$-free graphs of bounded clique-width and one of unbounded clique-width. The four new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of clique-width for the first three cases we develop a technique based on bounding …
If a graph has no induced subgraph isomorphic to any graph in a finite family $\{H_1,\ldots,H_p\}$, it is said to be $(H_1,\ldots,H_p)$-free. The class of $H$-free graphs has bounded clique-width if and only if $H$ is an induced subgraph of the 4-vertex path $P_4$. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs $H_1$ and $H_2$. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded cli…
Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates several studies to investigate whether the clique-width of graphs in a certain class is bounded or not. We focus on unigraphs, that is, graphs that are uniquely determined by their degree sequences up to isomorphism. We show that every unigraph has clique-width at most 4. It follows that many problems that are NP-hard in general are polynomial-time solvable for unigraphs.
We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width …
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$cycle covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$cycle. The minimum weight of $k-$cycle covering is the weighted covering number on $k-$cycle, denoted by $τ_{k}(G_{w})$. In this paper, we design a $k-1/2$ approximation algorithm for the weighted covering number on $k-$cycle when $k$ is odd. Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$clique covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$clique. The minimum weight of $k-$cl…
Clique-width is a well-known graph parameter. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded clique-width. The same holds for NLC-width. In this paper we study the behavior of clique-width and NLC-width under various graph operations and graph transformations. We give upper and lower bounds for the clique-width and NLC-width of the modified graphs in terms of the clique-width and NLC-width of the involved graphs.
Multi-clique-width is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing $c$-colorability. In particular, $c$-colorability can be tested in time linear in $n$ and singly exponential in $c$ and the width $k$ of a given multi-$k$-expression. For these tasks, the running time as a function…
We investigate a new width parameter, the fusion-width of a graph. It is a natural generalization of the tree-width, yet strong enough that not only graphs of bounded tree-width, but also graphs of bounded clique-width, trivially have bounded fusion-width. In particular, there is no exponential growth between tree-width and fusion-width, as is the case between tree-width and clique-width. The new parameter gives a good intuition about the relationship between tree-width and clique-width.
Clique-width is a graph invariant that has been widely studied in combinatorics and computer science. However, computing the clique-width of a graph is an intricate problem, the exact clique-width is not known even for very small graphs. We present a new method for computing the clique-width of graphs based on an encoding to propositional satisfiability (SAT) which is then evaluated by a SAT solver. Our encoding is based on a reformulation of clique-width in terms of partitions that utilizes an efficient encoding of cardinality constraints. Our SAT-based method is the first to discover the exa…
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…
We continue the study initiated by Bonomo-Braberman and Gonzalez in 2020 on $r$-locally checkable problems. We propose a dynamic programming algorithm that takes as input a graph with an associated clique-width expression and solves a $1$-locally checkable problem under certain restrictions. We show that it runs in polynomial time in graphs of bounded clique-width, when the number of colors of the locally checkable problem is fixed. Furthermore, we present a first extension of our framework to global properties by taking into account the sizes of the color classes, and consequently enlarge the…
Clique-width is a well-studied graph parameter owing to its use in understanding algorithmic tractability: if the clique-width of a graph class ${\cal G}$ is bounded by a constant, a wide range of problems that are NP-complete in general can be shown to be polynomial-time solvable on ${\cal G}$. For this reason, the boundedness or unboundedness of clique-width has been investigated and determined for many graph classes. We survey these results for hereditary graph classes, which are the graph classes closed under taking induced subgraphs. We then discuss the algorithmic consequences of these r…
We prove that edge contractions do not preserve the property that a set of graphs has bounded clique-width. This property is preserved by contractions of edges, one end of which is a vertex of degree 2.
Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width $k$ given with a $k$-expression, Dominating Set can be solved in $4^k n^{O(1)}$ time. However, no FPT algorithm is known for computing an optimal $k$-expression. For a graph of clique-width $k$, if we rely on known algorithms to compute a $(2^{3k}-1)$-expression via rank-width and then solving Dominating Set using the $(2^{3k}-1)$-expression, the above algorithm will only give a runtime of $4^{2^{3k}} n^{O(1)}$. There have been results which overcome this exponential jump; the best …
Durham University（杜伦大学）；Charles University, Prague（布拉格查理大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Simon Fraser University（西蒙菲莎大学）；Warsaw University of Technology（华沙理工大学）
We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has $10$ vertices. We determine the complete set of $10$-vertex and $11$-vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order $13$, as does the smal…
Indeterminate strings have received considerable attention in the recent past; see for example Christodoulakis et al 2015 and Helling et al 2017. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper solves a related problem proposed in Helling et al 2017: compute $Θ_n(m)$, which is the size of the largest possible mi…
Daligault, Rao and Thomassé asked whether a hereditary class of graphs well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph $H$, as such graphs are well-quasi-ordered and are of bounded cli…
We give an algorithm that, for every fixed k, decides isomorphism of graphs of rank width at most k in polynomial time. As the clique width of a graph is bounded in terms of its rank width, we also obtain a polynomial time isomorphism test for graph classes of bounded clique width.
We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm f…
Daligault, Rao and Thomassé asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether the question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs $H_1,H_2$ are forbidden. We confirm it for…
In this paper, we provide polynomial-time algorithms for different extensions of the matching counting problem, namely maximal matchings, path matchings (linear forest) and paths, on graph classes of bounded clique-width. For maximal matchings, we introduce matching-cover pairs to efficiently handle maximality in the local structure, and develop a polynomial time algorithm. For path matchings, we develop a way to classify the path matchings in a polynomial number of equivalent classes. Using these, we develop dynamic programing algorithms that run in polynomial time of the graph size, but in e…
We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth and of bounded cliquewidth. It also implies that monadically stable classes of bounded twin-widthare linearly $χ$-bounded.
The {\it clique cover width} of $G$, denoted by $ccw(G)$, is the minimum value of the bandwidth of all graphs that are obtained by contracting the cliques in a clique cover of $G$ into a single vertex. For $i=1,2,...,d,$ let $G_i$ be a graph with $V(G_i)=V$, and let $G$ be a graph with $V(G)=V$ and $E(G)=\cap_{i=1}^d(G_i)$, then we write $G=\cap_{i=1}^dG_i$ and call each $G_i,i=1,2,...,d$ a factor of $G$. We are interested in the case where $G_1$ is chordal, and $ccw(G_i),i=2,3...,d$ for each factor $G_i$ is "small". Here we show a negative result. Specifically, let ${\hat G}(k,n)$ be the grap…
In this paper we compare and illustrate the algorithmic use of graphs of bounded tree-width and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-width in polynomial time. We also present linear time algorithms for computing the latter four basic graph parameters on trees, i.e. graphs of tree-width 1, and on co-graphs, i.e. graphs of clique-width at …
Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}\cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a $2^{O(k)}\cdot n$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(k\cdot \log(k))}\cdot n^{O(1)}$ or worse.
The Colouring problem is that of deciding, given a graph $G$ and an integer $k$, whether $G$ admits a (proper) $k$-colouring. For all graphs $H$ up to five vertices, we classify the computational complexity of Colouring for $(\mbox{diamond},H)$-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for $(\mbox{diamond}, P_1+2P_2)$-free graphs. Our technique for handling this case is to reduce the graph under consideration to a $k$-partite graph that has a very specific decomposition. As a by-product of this general technique we are als…
The Target Set Selection problem takes as an input a graph $G$ and a non-negative integer threshold $ \mbox{thr}(v) $ for every vertex $v$. A vertex $v$ can get active as soon as at least $ \mbox{thr}(v) $ of its neighbors have been activated. The objective is to select a smallest possible initial set of vertices, the target set, whose activation eventually leads to the activation of all vertices in the graph. We show that Target Set Selection is in FPT when parameterized with the combined parameters clique-width of the graph and the maximum threshold value. This generalizes all previous FPT-m…
We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$ for all but six pairs $(H_1,H_2)$. Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open ca…
We give an $O(n^4)$ algorithm to find a minimum clique cover of a (bull, $C_4$)-free graph, or equivalently, a minimum colouring of a (bull, $2K_2$)-free graph, where $n$ is the number of vertices of the graphs.
A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs. In this paper we obtain several structural and algori…
The biclique cover number $(\text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $\text{bc}(G) \geq \lceil \log_2(\text{mc}(G^c)) \rceil \geq \lceil \log_2(χ(G)) \rceil$ for an arbitrary graph $G$, where $χ(G)$ is the chromatic number of $G$ and $\text{mc}(G^c)$ is the number of maximal cliques of the complementary graph $G^c$, i.e., the number of maximal independent sets of $G$. We also show that $\lceil \log_2(\text{mc}(G^c)) \rceil$ could be a strictly tighter lower bound of the bicl…
Let $α(G)$ and $β(G)$, denote the size of a largest independent set and the clique cover number of an undirected graph $G$. Let $H$ be an interval graph with $V(G)=V(H)$ and $E(G)\subseteq E(H)$, and let $ϕ(G,H)$ denote the maximum of ${β(G[W])\over α(G[W])}$ overall induced subgraphs $G[W]$ of $G$ that are cliques in $H$. The main result of this paper is to prove that for any graph $G$ $${β(G)}\le 2 α(H)ϕ(G,H)(\log α(H)+1),$$ where, $α(H)$ is the size of a largest independent set in $H$. We further provide a generalization that significantly unifies or improves some past algorithmic and struc…
An abundance of real-world problems manifest as covering edges and/or vertices of a graph with cliques that are optimized for some objectives. We consider different structural parameters of graph, and design fixed-parameter tractable algorithms for a number of clique cover problems. Using a set representation of graph, we introduce a framework for computing clique cover with different objectives. We demonstrate use of the framework for a variety of clique cover problems. Our results include a number of new algorithms with exponential to double exponential improvements in the running time.
In an attempt to find a polynomial-time algorithm for the edge-clique cover problem on cographs we tried to prove that the edge-clique graphs of cographs have bounded rankwidth. However, this is not the case. In this note we show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth.
Structural parameters of graph (such as degeneracy and arboricity) had rarely been considered when designing algorithms for $\textit{(edge) clique cover}$ problems. Taking degeneracy of graph into account, we present a greedy framework and two fixed-parameter tractable algorithms for $\textit{clique cover}$ problems. We introduce a set theoretic concept and demonstrate its use in the computations of different objectives of $\textit{clique cover}$. Furthermore, we show efficacy of our algorithms in practice.
Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Wil…
A graph is called $P_t$-free} if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geq 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. I…
Given two graphs $G$ and $H$, we say that $G$ contains $H$ as an induced minor if a graph isomorphic to $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. We study the complexity of Graph Isomorphism on graphs that exclude a fixed graph as an induced minor. More precisely, we determine for every graph $H$ that Graph Isomorphism is polynomial-time solvable on $H$-induced-minor-free graphs or that it is GI-complete. Additionally, we classify those graphs $H$ for which $H$-induced-minor-free graphs have bounded clique-width. These two results complement similar …
We show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs. We show that the independent set problem on edge-clique graphs of cographs and of distance-hereditary graphs can be solved in O(n^4) time. We show that the independent set problem on edge-clique graphs of graphs without odd wheels remains NP-complete.
A "clique minor" in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The "Hadwiger number" h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the Cartesian product G*H. Our main result is a rough structural characterisation theorem for Cartesian products with bounded Hadwiger number. It implies that if the product of two sufficiently large graphs has bounded Hadwiger number then it is one of the following graphs: - a planar grid with a vortex of bounded width in the outerface, - a cylindr…
