Cocomparability Graphs Research Articles

Bipartite Analogues of Comparability and Cocomparability Graphs

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 24 May 2019Accepted: 30 June 2020Published online: 15 September 2020Keywordscocomparability bigraph, chordal bigraph, interval bigraph, interval containment bigraph, two-directional orthogonal-ray graph, characterization, orientation, vertex ordering, invertible pair, asteroid, edge-asteroid, recognition, polynomial time algorithmAMS Subject Headings05C75, 05C85Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

A recognition algorithm for simple-triangle graphs

A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a longstanding open problem, which was recently settled. This paper provides a new recognition algorithm for simple-triangle graphs to improve the time bound from O(n2m¯) to O(nm), where n, m, and m¯ are the number of vertices, edges, and non-edges of the graph, respectively. The algorithm uses the vertex ordering characterization that a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices containing both an alternating orientation of the graph and a transitive orientation of the complement of the graph. We also show, as a byproduct, that an alternating orientation can be obtained in O(nm) time for cocomparability graphs, and it is NP-complete to decide whether a graph has an orientation that is alternating and acyclic.

Classes of Graphs with $e$-Positive Chromatic Symmetric Function

In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were $e$-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are $e$-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are $e$-positive. We introduce the concept of strongly $e$-positive to denote a graph whose induced subgraphs are all $e$-positive, and conjecture that a graph is strongly $e$-positive if and only if it is (claw, net)-free.

Bounds on the Bend Number of Split and Cocomparability Graphs

A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A k-bend path is a path made up of at most k + 1 line segments. A Bk-VPG representation of a graph is a collection of k-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a Bk-VPG representation are calledBk-VPG graphs. The bend number of a graph G, denoted by bend(G), is the minimum integer k for which G has a Bk-VPG representation. In this paper, we study the relationship between poset dimension and bend number of cocomparability graphs. It is known that the poset dimension dim(G) of a cocomparability graph G is greater than or equal to its bend number bend(G). Cohen et al. (order2016) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each $m, t \in \mathbb {N}$ , there exists a cocomparability graph Gt, m with t m. The techniques used to prove the above result allow us to partially address the open question posed by Chaplick et al. (wg2012) who asked whether Bk-VPG-chordal $\subsetneq ~B_{k + 1}$ -VPG-chordal for all $k \in \mathbb {N}$ . We address this by proving that there are infinitely many $m \in \mathbb {N}$ such that Bm-VPG-split $\subsetneq ~B_{m + 1}$ -VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all $t \in \mathbb {N}$ , Bt-VPG- $Forb(C_{\geq 5})~\subsetneq ~B_{4t + 29}$ -VPG-Forb(C≥ 5), where Forb(C≥ 5) denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all $t \in \mathbb {N}$ , PBt-VPG-split $\subsetneq PB_{36t + 80}$ -VPG-split, where PBt-VPG denotes the class of graphs with proper bend number at most t (i.e. it has a Bt-VPG representation in which two paths have only finitely many intersection points, each intersection point belongs to exactly two paths, and whenever two paths intersect they cross each other).

Maximum Induced Matching Algorithms via Vertex Ordering Characterizations

We study the maximum induced matching problem on a graph G. Induced matchings correspond to independent sets in $$L^2(G)$$, the square of the line graph of G. The problem is NP-complete on bipartite graphs. In this work, we show that for a number of graph families characterized by vertex orderings, almost all forbidden patterns on three vertices are preserved when taking the square of the line graph. These orderings can be computed in linear time in the size of the input graph. In particular, given $$\mathcal {G}$$ a graph class, and a graph $$G=(V,E) \in \mathcal {G}$$ with a corresponding vertex ordering $$\sigma $$ of V, one can produce (in linear time in the size of G) an ordering on the vertices of $$L^2(G)$$, that shows that $$L^2(G) \in \mathcal {G}$$ without computing the line graph or the square of the line graph of G. These results generalize and unify previous ones on showing closure under $$L^2(\cdot )$$ for various graph families. Furthermore, these orderings on $$L^2(G)$$ can be exploited algorithmically to compute a maximum induced matching on G faster. We illustrate this latter fact in the second half of the paper where we focus on cocomparability graphs, a large graph class that includes interval, permutation, trapezoid graphs, and co-graphs, and we present the first $$\mathcal {O}(mn)$$ time algorithm to compute a maximum weighted induced matching on cocomparability graphs; an improvement from the best known $$\mathcal {O}(n^4)$$ time algorithm for the unweighted case.

Counting independent sets in cocomparability graphs

We show that the number of independent sets in cocomparability graphs can be counted in linear time, as can counting cliques in comparability graphs. By contrast, counting cliques in cocomparability graphs and counting independent sets in comparability graphs are #P-complete. We extend these results to counting maximal cliques and independent sets. We also consider the fixed-parameter versions of counting cliques and independent sets of given size k. Finally, we combine the results to show that both counting cliques and independent sets in permutation graphs are in linear time.

Reconfiguration of colorable sets in classes of perfect graphs

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (called a reconfiguration sequence) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

Perfect Phylogenies via Branchings in Acyclic Digraphs and a Generalization of Dilworth’s Theorem

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, 2018) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurović et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs. We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including (1) a strengthening of the heuristic by Hujdurović et al. via a new min-max result in digraphs generalizing Dilworth’s theorem, which may be of independent interest; (2) APX-hardness results for both problems; (3) approximation algorithms; and (4) exponential-time algorithms solving the two problems to optimality faster than the naïve brute-force approach. Our work relates to several well-studied notions in combinatorial optimization: chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.

A Linear-Time Algorithm for Maximum-Cardinality Matching on Cocomparability Graphs

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For general $m$-edge and $n$-vertex graphs, it is well known to be solvable in $O(m\sqrt{n})$ time. We present a linear-time algorithm to find maximum-cardinality matchings on cocomparability graphs, a prominent subclass of perfect graphs that strictly contains interval graphs as well as permutation graphs. Our greedy algorithm is based on the recently discovered Lexicographic Depth First Search (LDFS).

A width parameter useful for chordal and co-comparability graphs

Belmonte and Vatshelle (TCS 2013) used mim-width, a graph width parameter bounded on interval graphs and permutation graphs, to explain existing algorithms for many domination-type problems on those graph classes. We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs Kt⊟Kt and Kt⊟St are graphs obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that:•interval graphs are (K3⊟S3)-free chordal graphs; and (Kt⊟St)-free chordal graphs have mim-width at most t−1,•permutation graphs are (K3⊟K3)-free co-comparability graphs; and (Kt⊟Kt)-free co-comparability graphs have mim-width at most t−1,•chordal graphs and co-comparability graphs have unbounded mim-width in general. We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time nO(t) on (Kt⊟St)-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless P=NP.We generalize these ideas to bigger graph classes. We introduce a new width parameter sim-width, of stronger modeling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding Kt⊟Kt and Kt⊟St as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width.

Lower bounds on the mim-width of some graph classes

mim-width is a recent graph width measure that has seen applications in graph algorithms and problems related to propositional satisfiability. In this paper, we show linear lower bounds for the mim-width of strongly chordal split graphs, co-comparability graphs and circle graphs. This improves and refines lower bounds that were known before, some of them only conditionally. In the case of strongly chordal graphs not even a conditional lower bound was known before. All of the bounds given are optimal up to constants.

Complexity and Algorithms for Finding a Perfect Phylogeny from Mixed Tumor Samples.

Hajirasouliha and Raphael (WABI 2014) proposed a model for deconvoluting mixed tumor samples measured from a collection of high-throughput sequencing reads. This is related to understanding tumor evolution and critical cancer mutations. In short, their formulation asks to split each row of a binary matrix so that the resulting matrix corresponds to a perfect phylogeny and has the minimum number of rows among all matrices with this property. In this paper, we disprove several claims about this problem, including an NP-hardness proof of it. However, we show that the problem is indeed NP-hard, by providing a different proof. We also prove NP-completeness of a variant of this problem proposed in the same paper. On the positive side, we propose an efficient (though not necessarily optimal) heuristic algorithm based on coloring co-comparability graphs, and a polynomial time algorithm for solving the problem optimally on matrix instances in which no column is contained in both columns of a pair of conflicting columns. Implementations of these algorithms are freely available at https://github.com/alexandrutomescu/MixedPerfectPhylogeny.

On the Computational Complexity of Vertex Integrity and Component Order Connectivity

The Weighted Vertex Integrity (wVI) problem takes as input an n-vertex graph G, a weight function $$w:V(G)\rightarrow {\mathbb {N}}$$w:V(G)źN, and an integer p. The task is to decide if there exists a set $$X\subseteq V(G)$$X⊆V(G) such that the weight of X plus the weight of a heaviest component of $$G-X$$G-X is at most p. Among other results, we prove that:(1)wVI is NP-complete on co-bipartite graphs, even if each vertex has weight 1;(2)wVI can be solved in $$O(p^{p+1}n)$$O(pp+1n) time;(3)wVI admits a kernel with at most $$p^3$$p3 vertices. Result (1) refutes a conjecture by Ray and Deogun (J Comb Math Comb Comput 16:65---73, 1994) and answers an open question by Ray et al. (Ars Comb 79:77---95, 2006). It also complements a result by Kratsch et al. (Discret Appl Math 77(3):259---270, 1997), stating that the unweighted version of the problem can be solved in polynomial time on co-comparability graphs of bounded dimension, provided that an intersection model of the input graph is given as part of the input. An instance of the Weighted Component Order Connectivity (wCOC) problem consists of an n-vertex graph G, a weight function $$w:V(G)\rightarrow {\mathbb {N}}$$w:V(G)źN, and two integers k and $$\ell $$l, and the task is to decide if there exists a set $$X\subseteq V(G)$$X⊆V(G) such that the weight of X is at most k and the weight of a heaviest component of $$G-X$$G-X is at most $$\ell $$l. In some sense, the wCOC problem can be seen as a refined version of the wVI problem. We obtain several classical and parameterized complexity results on the wCOC problem, uncovering interesting similarities and differences between wCOC and wVI. We prove, among other results, that:(4)wCOC can be solved in $$O(\min \{k,\ell \}\cdot n^3)$$O(min{k,l}·n3) time on interval graphs, while the unweighted version can be solved in $$O(n^2)$$O(n2) time on this graph class;(5)wCOC is W[1]-hard on split graphs when parameterized by k or by $$\ell $$l;(6)wCOC can be solved in $$2^{O(k\log \ell )} n$$2O(klogl)n time;(7)wCOC admits a kernel with at most $$k\ell (k+\ell )+k$$kl(k+l)+k vertices. We also show that result (6) is essentially tight by proving that wCOC cannot be solved in $$2^{o(k \log \ell )}n^{O(1)}$$2o(klogl)nO(1) time, even when restricted to split graphs, unless the Exponential Time Hypothesis fails.

On the Power of Graph Searching for Cocomparability Graphs

In this paper we study how graph searching on a cocomparability graph $G$ can be used to produce cocomp orderings (i.e., orderings that are linear extensions of some transitive orientation of $\overline{G}$) that yield simple algorithms for various intractable problems in general. Such techniques have been used to find a simple certifying algorithm for the minimum path cover problem. In particular we present a characterization of the searches that preserve cocomp orderings when used as a “$^+$” sweep. This allows us to present a toolbox of different graph searches and a framework to solve various problems on cocomparability graphs. We illustrate these techniques by describing a very simple certifying algorithm for the maximum independent set problem as well as a simple permutation graph recognition algorithm.

A linear time algorithm to compute a maximum weighted independent set on cocomparability graphs

The maximum weight independent set (WMIS) problem is a well-known NP-hard problem. It is a generalization of the maximum cardinality independent set problem where all the vertices have identical weights. There is an O(n2) time algorithm to compute a WMIS for cocomparability graphs by computing a maximum weight clique on the corresponding complement of the graph [1]. We present the first O(m+n) time algorithm to compute a WMIS directly on the given cocomparability graph, where m and n are the number of edges and vertices of the graph respectively. As a corollary, we get the minimum weight vertex cover of a cocomparability graph in linear time as well.

A new LBFS-based algorithm for cocomparability graph recognition

How much can you learn about the structure of a given graph, using a series of successive graph searches? We consider here cocomparability graphs which are the complement of the graphs that admit a transitive orientation (comparability graphs). We study the use of a series of successive LBFS searches in order to capture the structure of these graphs. On cocomparability graphs many problems like coloring, Hamilton path, maximal clique which are NP-complete in the general case can be solved in polynomial time. For all those problems a decisive step is to find a cocomp ordering. We will prove the correctness of an algorithm to find a cocomp ordering using |V(G)| successive LBFS. Our main result answers a question asked in Corneil, Olariu and Stewart [2009] about the existence of a multisweep LBFS algorithm for cocomparability graphs, and lays the foundation of a simple linear time multisweep LBFS algorithm to find a cocomp ordering.

On graphs associated to sets of rankings

In this paper we analyze families of rankings by studying structural properties of graphs. Given a finite number of elements and a set of rankings of those elements, two elements compete when they exchange their relative positions in at least two rankings, and we can associate an undirected graph to a set of rankings by connecting elements that compete. We call this graph a competitivity graph. Competitivity graphs have already appeared in the literature as co-comparability graphs, f-graphs or intersection graphs associated to a concatenation of permutation diagrams. We introduce certain important sets of nodes in a competitivity graph. For example, nodes that compete among them form a competitivity set and nodes connected by chains of competitors form a set of eventual competitors. These sets are analyzed and a method to obtain sets of eventual competitors directly from a set of rankings is shown.

The hub number of co-comparability graphs

A set H⊆V is a hub set of a graph G=(V,E) if, for every pair of vertices u,v∈V∖H, either u is adjacent to v or there exists a path from u to v such that all intermediate vertices are in H. The hub number of G, denoted by h(G), is the minimum size of a hub set in G. The connected hub number of G, denoted by hc(G), is the minimum size of a connected hub set in G. In this paper, we prove that h(G)=hc(G) for co-comparability graphs G and characterize the case for which γc(G)=hc(G) in this class of graphs, where γc(G) denotes the connected domination number of G. We also show that h(G) can be computed in O(|V|) time for trapezoid graphs and in O(|V|3) time for co-comparability graphs.

On the intersection of tolerance and cocomparability graphs

Tolerance graphs have been extensively studied since their introduction, due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured by Golumbic, Monma, and Trotter in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocomparability graphs can be efficiently recognized, a positive answer to this conjecture in the general case would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize each of these classes of graphs separately. This longstanding conjecture has been proved under some– rather strong –structural assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs. Our main result in this article is that the above conjecture is true for every graph G that admits a tolerance representation with exactly one unbounded vertex; note that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. Moreover, our results imply as a corollary that the conjecture of Golumbic, Monma, and Trotter is true for every graph G=(V,E) that has no three independent vertices a,b,c∈V such that N(a)⊂N(b)⊂N(c), where N(v) denotes the set of neighbors of a vertex v∈V; this is satisfied in particular when G is the complement of a triangle-free graph (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are constructive, in the sense that, given a tolerance representation R of a graph G, we transform R into a bounded tolerance representation R∗ of G. Furthermore, we conjecture that any minimal tolerance graph G that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic, Monma, and Trotter, it suffices to prove our conjecture.

Vertex Ordering Characterizations of Graphs of Bounded Asteroidal Number

Asteroidal Triple-free (AT-free) graphs have received considerable attention due to their inclusion of various important graphs families, such as interval and cocomparability graphs. The asteroidal number of a graph is the size of a largest subset of vertices such that the removal of the closed neighborhood of any vertex in the set leaves the remaining vertices of the set in the same connected component. (AT-free graphs have asteroidal number at most 2.) In this article, we characterize graphs of bounded asteroidal number by means of a vertex elimination ordering, thereby solving a long-standing open question in algorithmic graph theory. Similar characterizations are known for chordal, interval, and cocomparability graphs.