Cocomparability Graph Research Articles

Latest Algorithms on Particular Graph Classes

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

Coloring squares of graphs via vertex orderings

We provide upper bounds on the chromatic number of the square of graphs, which have vertex ordering characterizations. We prove that [Formula: see text] is [Formula: see text]-colorable when [Formula: see text] is a cocomparability graph, [Formula: see text]-colorable when [Formula: see text] is a strongly orderable graph and [Formula: see text]-colorable when [Formula: see text] is a dually chordal graph, where [Formula: see text] is the maximum degree and [Formula: see text] = max[Formula: see text] is the multiplicity of the graph [Formula: see text]. This improves the currently known upper bounds on the chromatic number of squares of graphs from these classes.

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).

Robinsonian Matrices: Recognition Challenges

Ultrametric inequality is involved in different operations on (dis)similarity matrices. Its coupling with a compatible ordering leads to nice interpretations in seriation problems. We accurately review the interval graph recognition problem for its tight connection with recognizing a dense Robinsonian dissimilarity (precisely, in the anti-ultrametric case). Since real life matrices are prone to errors or missing entries, we address the sparse case and make progress towards recognizing sparse Robinsonian dissimilarities with lexicographic breadth first search. The ultrametric inequality is considered from the same graph point of view and the intimate connection between cocomparability graph and dense Robinsonian similarity is established. The current trend in recognizing special graph structures is examined in regard to multiple lexicographic search sweeps. Teaching examples illustrate the issues addressed for both dense and sparse symmetric matrices.

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.

Posets and VPG Graphs

We investigate the class of intersection graphs of paths on a grid (VPG graphs), and specifically the relationship between the bending number of a cocomparability graph and the poset dimension of its complement. We show that the bending number of a cocomparability graph G is at most the poset dimension of the complement of G minus one. Then, via Ramsey type arguments, we show our upper bound is best possible.

Thin strip graphs

A unit disk graph is a c-strip graph if it has a unit disk representation in which all centers of the unit disks lie between two parallel lines at distance c. The classes of c-strip graphs for various c are studied in the literature. For example, the 0-strip graphs are exactly the unit interval graphs, and every 3/2-strip graph is a co-comparability graph. In this paper, we introduce the class of thin strip graphs and study their properties. A graph is a thin strip graph if it is a c-strip graph for every c>0. We show that there is no constant t such that the t-strip graphs are exactly the thin strip graphs. We also show that the class of thin strip graphs properly includes the class of mixed unit interval graphs.

Task scheduling for directed cyclic graph using matching technique

The scheduling and mapping of task graph to processors is considered to be the most crucial NP-complete in parallel and distributed computing systems. In this paper, the theoretical graph application using matching is presented to assign a number of tasks onto two processors. This paper addresses a directed-weighted cyclic graph. The effort is to reduce the graph onto directed acyclic graph. A co-comparability graph is presented in order to assign the task onto two processors. Combining several innovative techniques lead to an efficient graph-mapping concept, called DCGSimplify. Our simulation model found that the proposed techniques and algorithms are easy to be implemented.

LDFS-Based Certifying Algorithm for the Minimum Path Cover Problem on Cocomparability Graphs

For graph $G(V,E)$, a minimum path cover (MPC) is a minimum cardinality set of vertex disjoint paths that cover $V$ (i.e., every vertex of $G$ is in exactly one path in the cover). This problem is a natural generalization of the Hamiltonian path problem. Cocomparability graphs (the complements of graphs that have an acyclic transitive orientation of their edge sets) are a well studied subfamily of perfect graphs that includes many popular families of graphs such as interval, permutation, and cographs. Furthermore, for every cocomparability graph $G$ and acyclic transitive orientation of the edges of $\overline{G}$ there is a corresponding poset $P_G$; it is easy to see that an MPC of $G$ is a linear extension of $P_G$ that minimizes the bump number of $P_G$. Although there are directly graph-theoretical MPC algorithms (i.e., algorithms that do not rely on poset formulations) for various subfamilies of cocomparability graphs, notably interval graphs, until now all MPC algorithms for cocomparability graphs themselves have been based on the bump number algorithms for posets. In this paper we present the first directly graph-theoretical MPC algorithm for cocomparability graphs; this algorithm is based on two consecutive graph searches followed by a certifying algorithm. Surprisingly, except for a lexicographic depth first search (LDFS) preprocessing step, this algorithm is identical to the corresponding algorithm for interval graphs. The running time of the algorithm is $O({\rm min}(n^2, n + {\rm mloglogn}))$, with the nonlinearity coming from LDFS.

Collective tree spanners of graphs

In this paper we introduce a new notion of collective tree spanners. We say that a graph G=(V,E)admits a system of $\mu$ collective additive tree r-spanners if there is a system T(G) of at most $\mu$ spanning trees of G such that for any two vertices x,y of G a spanning tree T\in \cT(G) exists such that d_T(x,y)\leq d_G(x,y)+r. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log2n collective additive tree 2-spanners. These results are complemented by lower bounds, which say that any system of collective additive tree 1-spanners must have $\Omega(\sqrt{n})$ spanning trees for some chordal graphs and $\Omega(n)$ spanning trees for some chordal bipartite graphs and some cocomparability graphs. Furthermore, we show that any c-chordal graph admits a system of at most log2n collective additive tree (2\lfloor c/2\rfloor)-spanners, any circular-arc graph admits a system of two collective additive tree 2-spanners. Towards establishing these results, we present a general property for graphs, called (\al,r)$-decomposition, and show that any $(\al,r)$-decomposable graph G with n vertices admits a system of at most $\log_{1/\al} n$ collective additive tree $2r$-spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs. For any graph on n vertices admitting a system of at most $\mu$ collective additive tree r-spanners, there is a routing scheme of deviation r with addresses and routing tables of size $O(\mu \log^2n/\log \log n)$ bits per vertex. This leads, for example, to a routing scheme of deviation $(2\lfloor c/2\rfloor)$ with addresses and routing tables of size $O(\log^3n/\log \log n)$ bits per vertex on the class of c-chordal graphs.

On the minimum number of processors for scheduling problems with communicationdelays

Scheduling problems with interprocessor communication delays are recent problemsarising with the development of new message‐passing architectures whose number of processorsis increasing more and more. The scheduling problem with communication delays isNP‐complete even on an unlimited number of processors, and most of the restrictions whichare known to make the problem easy suppose that the number of processors is unlimited.The aim of this paper is to estimate the minimum number of processors required to realizea schedule that minimizes the makespan for problems with communication delays. Contraryto problems without communication costs, we show that the minimum number of partitioningpaths of tasks is not an upper bound. Then we propose two upper bounds b and b whichare valid independent of task processing times and communication costs. We show thatb can be calculated in O(n 2 ) if n designates the number of tasks. Then we give an algorithmfor determining b in the special case when the precedence graph is an in‐tree or anout-tree. A specific study of SCT task systems (Small Communication Time) shows that theachromatic number of the cocomparability graph is an upper bound on the minimum numberof processors.

Weighted domination of cocomparability graphs

It is shown in this paper that the weighted domination problem and its three variants, the weighted connected domination, total domination, and dominating clique problems are NP-complete on cobipartite graphs when arbitrary integer vertex weights are allowed and all of them can be solved in polynomial time on cocomparability graphs if vertex weights are integers and less than or equal to a constant c. The results are interesting because cocomparability graphs properly contain cobipartite graphs and the cardinality cases of the above problems are trivial on cobipartite graphs. On the other hand, an O(¦V¦ 2) algorithm is given for the weighted independent perfect domination problem of a cocomparability graph G = ( V. E).

1-Tough cocomparability graphs are hamiltonian

We show that every 1-tough cocomparability graph has a Hamilton cycle. This settles a conjecture of Chvàtal for the case of cocomparability graphs. Our approach is based on exploiting the close relationship of the problem to the scattering number and the path cover number.

An NC algorithm for the clique cover problem in cocomparability graphs and its application

In this paper, we present an extremely simple NC algorithm for finding a minimum clique cover in a cocomparability graph. Our algorithm takes O( log n) time using O( n 3 + ε ) processors on the CRCW PRAM, where n is the number of vertices. It is also shown that this algorithm can be applied to solve in parallel the depth-first search problem for permutation graphs. The best known DFS algorithm for the permutation graphs runs in O( log 2 n) time on a CRCW PRAM model.

Polynomial Algorithms for Hamiltonian Cycle in Cocomparability Graphs

Finding a Hamiltonian cycle in a graph is one of the classical NP-complete problems. Complexity of the Hamiltonian problem in permutation graphs has been a well-known open problem. In this paper the authors settle the complexity of the Hamiltonian problem in the more general class of cocomparability graphs. It is shown that the Hamiltonian cycle existence problem for cocomparability graphs is in P. A polynomial time algorithm for constructing a Hamiltonian path and cycle is also presented. The approach is based on exploiting the relationship between the Hamiltonian problem in a cocomparability graph and the bump number problem in a partial order corresponding to the transitive orientation of its complementary graph.

Treewidth of cocomparability graphs and a new order-theoretic parameter

We show that the pathwidth of a cocomparability graphG equals its treewidth. The proof is based on a new notion, calledinterval width, for a partial orderP, which is the smallest width of an interval order contained inP, and which is shown to be equal to the treewidth of its cocomparability graph. We observe also that determining any of these parameters isNP-hard and we establish some connections between classical dimension notions of partial orders and interval width. In fact we develop approximation algorithms for interval width ofP whose performance ratios depend on the dimension and interval dimension ofP, respectively.

On-Line Coloring and Recursive Graph Theory

An on-line vertex coloring algorithm receives the vertices ofa graph in some externally determined order, and, whenever a new vertex is presented, the algorithm also learns to which of the previously presented vertices the new vertex is adjacent. As each vertex is received, the algorithm must make an irrevocable choice of a color to assign the new vertex, and it makes this choice without knowledge of future vertices. A class of graphs $r$ is said to be on-line $\chi$-bounded if there exists an on-line algorithm $A$ and a function $f$ such that $A$ uses at most $f(\omega(G))$ colors to properly color any graph $G$ in \Gamma. If $H$ is a graph, let Forb($H$) denote the class of graphs that do not induce $H$. The goal of this paper is to establish that Forb($T$) is on-line $\chi$-bounded for every radius-2 tree $T$. As a corollary, the authors answer a question of Schmerl's; the authors show that every recursive cocomparability graph can be recursively colored with a number of colors that depends only on its clique number.

Hamiltonian cycle is polynomial on cocomparability graphs

Finding a Hamiltonian path or a Hamiltonian cycle in a general graph are classic NP-complete problems. In this paper we announce polynomial time solutions for these problems on cocomparability graphs. Our approach is based on exploiting the relationship between the Hamiltonian problem in a cocomparability graph G and the bump number problem in a partial order, the comparability graph of which is the complement of G.