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. In both cases, the algorithms rely on deep structural insights into the considered graph classes.
One of the main tools in the algorithms for MWIS in $P_5$-free graphs~[SODA 2014] and in $P_6$-free graphs~[SODA 2019] is the so-called Separator Covering Lemma that asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices. In this note we show that such a statement generalizes to $P_7$-free graphs and is false in $P_8$-free graphs. We also discuss analogues of such a statement for covering potential maximal cliques with unions of neighborhoods.
In a graph, a Clique-Stable Set separator (CS-separator) is a family $\mathcal{C}$ of cuts (bipartitions of the vertex set) such that for every clique $K$ and every stable set $S$ with $K \cap S = \emptyset$, there exists a cut $( W,W')$ in $\mathcal{C}$ such that $K \subseteq W$ and $S \subseteq W'$. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis [17] asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive …
Yannakakis' Clique versus Independent Set problem (CL-IS) in communication complexity asks for the minimum number of cuts separating cliques from stable sets in a graph, called CS-separator. Yannakakis provides a quasi-polynomial CS-separator, i.e. of size $O(n^{\log n})$, and addresses the problem of finding a polynomial CS-separator. This question is still open even for perfect graphs. We show that a polynomial CS-separator almost surely exists for random graphs. Besides, if H is a split graph (i.e. has a vertex-partition into a clique and a stable set) then there exists a constant $c_H$ for…
Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an {\em atom} if it has no clique separator. A {\em hole} is a chordless cycle with at least five vertices, and an {\em antihole} is the complement graph of a hole. A graph is {\em weakly chordal} if it is hole- and antihole-free. $K_4-e$ is also called {\em diamond}. {\em Paraglider} has five vertices four of which induce a diamond, and the fifth vertex sees exactly the two vertices of degree two in the diamond. In this paper we show that atoms of hole- and diamond-…
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of Haifa（海法大学）
Partitioning a graph using graph separators, and particularly clique separators, are well-known techniques to decompose a graph into smaller units which can be treated independently. It was previously known that the treewidth was bounded above by the sum of the size of the separator plus the treewidth of disjoint components, and this was obtained by the heuristic of filling in all edges of the separator making it into a clique. In this paper, we present a new, tighter upper bound on the treewidth of a graph obtained by only partially filling in the edges of a separator. In particular, the meth…
Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if $K\subseteq B$ and $S\subseteq W$. A Clique-Stable Set Separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique-Stable Set Separator cont…
Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard combinatorial optimization problems related to minimal triangulations, are broken into subproblems by block subgraphs defined by minimal separators. These ideas were expanded on by Bouchitté and Todinca, who used potential maximal cliques to solve these problems using a dynamic programming approach in time polynomial in the number of minimal separators of a graph. It is known that solutions to the perfect phylogeny problem, maximum compatibility problem, and unique perfect phylogeny problem are characterized by minimal …
Algorithm MLS (Maximal Label Search) is a graph search algorithm which generalizes algorithms MCS, LexBFS, LexDFS and MNS. On a chordal graph, MLS computes a peo (perfect elimination ordering) of the graph. We show how algorithm MLS can be modified to compute a pmo (perfect moplex ordering) as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and …
We show that 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. We show that the independent set problem on edge-clique graphs of graphs without odd wheels remains NP-complete. We present a PTAS for planar graphs and show that the problem is polynomial for planar graphs without triangle separators.
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…
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…
We consider the largest number of minimal separators a graph on n vertices can have at most. We give a new proof that this number is in $O( ((1+\sqrt{5})/2)^n n )$. We prove that this number is in $ω( 1.4521^n )$, improving on the previous best lower bound of $Ω(3^{n/3}) \subseteq ω( 1.4422^n )$. This gives also an improved lower bound on the number of potential maximal cliques in a graph. We would like to emphasize that our proofs are short, simple, and elementary.
Chordal graphs are the graphs in which every cycle of length at least four has a chord. A set $S$ is a vertex separator for vertices $a$ and $b$ if the removal of $S$ of the graph separates $a$ and $b$ into distinct connected components. A graph $G$ is chordal if and only if every minimal vertex separator is a clique. We study subclasses of chordal graphs defined by restrictions imposed on the intersections of its minimal separator cliques. Our goal is to characterize them by forbidden induced subgraphs. Some of these classes have already been studied such as chordal graphs in which two minima…
The atom graph of a graph is the graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in $O(min(n^α\log n, nm, n(n+\overline{m}))$ time, which is no more than the complexity of computing the atoms in the general case. %\par We extend our results to $α$-acyclic hypergraphs. We introduce the notion of union join graph, which is the union of all possible join trees; we apply our algorithms for a…
For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components. A vertex separator $S$ is minimal if it contains no other separator as a strict subset and a minimum vertex separator is a minimal vertex separator of least cardinality. A {\em clique} is a set of mutually adjacent vertices. A 2-tree is a connected graph in which every maximal clique is of size three and every minimal vertex separator is of size two. A spanning tree of a graph $G$ is a connected and an acyclic subgraph of $G$. In this paper, we focus our attention on two enumeration pr…
This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomposition. In a similar way, we introduce the class of switch cographs, the class of graphs that are totally decomposable w.r.t involution modular decomposition. This class generalizes the class of cogra…
In this paper a greedy algorithm to detect conflict cliques in interval graphs and circular-arc graphs is analyzed. In a graph, a stable set requires that at most one vertex is chosen for each edge. It is equivalent to requiring that at most one vertex for each maximal clique is chosen. We show that this algorithm finds all maximal cliques for interval graphs, i.e. it can compute the convex hull of the stable set polytope. In case of circular-arc graphs, the algorithm is not able to detect all maximal cliques, yet remaining correct. This problem occurs in the context of railway scheduling. A t…
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 …
Biró et al. (1992) introduced $H$-graphs, intersection graphs of connected subgraphs of a subdivision of a graph $H$. They are related to many classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. We negatively answer the 25-year-old question of Biró et al. which asks if $H$-graphs can be recognized in polynomial time, for a fixed graph $H$. We prove that it is NP-complete if $H$ contains the diamond graph as a minor. We provide a polynomial-time algorithm recognizing $T$-graphs, for each fixed tree $T$. When $T$ is a star $S_d$…
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.
We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ($cc$) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $cc$ is at most the number of taxa, in fractional hypertreewidth $cc$ is at most the number of hyperedges, and in treewidth of Bayesian networks $cc$ is at most the number of non-root nodes. We show that the number of minimal separators of graphs is…
Fractional graph isomorphism is the linear relaxation of an integer programming formulation of graph isomorphism. It preserves some invariants of graphs, like degree sequences and equitable partitions, but it does not preserve others like connectivity, clique and independence numbers, chromatic number, vertex and edge cover numbers, matching number, domination and total domination numbers. In this work, we extend the concept of fractional graph isomorphism to hypergraphs, and give an alternative characterization, analogous to one of those that are known for graphs. With this new concept we pro…
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 …
The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for odd-hole- and dart-free graphs as well…
We give alternative definitions for maximum matching width, e.g. a graph $G$ has $\operatorname{mmw}(G) \leq k$ if and only if it is a subgraph of a chordal graph $H$ and for every maximal clique $X$ of $H$ there exists $A,B,C \subseteq X$ with $A \cup B \cup C=X$ and $|A|,|B|,|C| \leq k$ such that any subset of $X$ that is a minimal separator of $H$ is a subset of either $A, B$ or $C$. Treewidth and branchwidth have alternative definitions through intersections of subtrees, where treewidth focuses on nodes and branchwidth focuses on edges. We show that mm-width combines both aspects, focusing…
