Permutation Graphs Research Articles

Edge-connectivity of permutation hypergraphs

In this note we provide a generalization of a result of Goddard et al. (2003) [4] on edge-connectivity of permutation graphs for hypergraphs. A permutation hypergraph Gπ is obtained from a hypergraph G by taking two disjoint copies of G and by adding a perfect matching between them. The main tool in the proof of the graph result was the theorem on partition constrained splitting off preserving k-edge-connectivity due to Bang-Jensen et al. (1999) [1]. Recently, this splitting off theorem was extended for hypergraphs by Bernáth et al. (accepted in Journal of Graph Theory) [2]. This extension made it possible to find a characterization of hypergraphs for which there exists a k-edge-connected permutation hypergraph.

Bounded coloring of co-comparability graphs and the pickup and delivery tour combination problem

The Double Traveling Salesman Problem with Multiple Stacks is a vehicle routing problem in which pickups and deliveries must be performed in two independent networks. The items are stored in stacks and repacking is not allowed. Given a pickup and a delivery tour, the problem of checking if there exists a valid distribution of items into s stacks of size h that is consistent with the given tours, is known as Pickup and Delivery Tour Combination (PDTC) problem.In the paper, we show that the PDTC problem can be solved in polynomial time when the number of stacks s is fixed but the size of each stack is not. We build upon the equivalence between the PDTC problem and the bounded coloring (BC) problem on permutation graphs: for the latter problem, s is the number of colors and h is the number of vertices that can get a same color. We show that the BC problem can be solved in polynomial time when s is a fixed constant on co-comparability graphs, a superclass of permutation graphs. To the contrary, the BC problem is known to be hard on permutation graphs when h≥6 is a fixed constant, but s is unbounded.

Erratum and addendum to “A linear time algorithm for finding all hinge vertices of a permutation graph” [Information Processing Letters 59 (2) (1996) 103–107

This note points out and corrects an error in the algorithm proposed in [Ting-Yem Ho, Yue-Li Wang and Ming-Tsan Juan, A linear time algorithm for finding all hinge vertices of a permutation graph, Information Processing Letters 59 (2) (1996) 103–107].

Edge search number of cographs

We give a linear-time algorithm for computing the edge search number of cographs, thereby resolving the computational complexity of edge searching on this graph class. To achieve this we give a characterization of the edge search number of the join of two graphs. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can all be computed efficiently. Furthermore, we are one step closer to computing the edge search number of permutation graphs.

Vertex splitting and the recognition of trapezoid graphs

Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the complement has a transitive orientation). The operation of “vertex splitting”, introduced in (Cheah and Corneil, 1996) [3], first augments a given graph G and then transforms the augmented graph by replacing each of the original graph’s vertices by a pair of new vertices. This “splitted graph” is a permutation graph with special properties if and only if G is a trapezoid graph. Recently vertex splitting has been used to show that the recognition problems for both tolerance and bounded tolerance graphs is NP-complete (Mertzios et al., 2010) [11]. Unfortunately, the vertex splitting trapezoid graph recognition algorithm presented in (Cheah and Corneil, 1996) [3] is not correct. In this paper, we present a new way of augmenting the given graph and using vertex splitting such that the resulting algorithm is simpler and faster than the one reported in (Cheah and Corneil, 1996) [3].

Computing minimum distortion embeddings into a path for bipartite permutation graphs and threshold graphs

The problem of computing minimum distortion embeddings of a given graph into a line (path) was introduced in 2004 and has quickly attracted significant attention with subsequent results appearing at recent stoc and soda conferences. So far, all such results concern approximation algorithms or exponential-time exact algorithms. We give the first polynomial-time algorithms for computing minimum distortion embeddings of graphs into a path when the input graphs belong to specific graph classes. In particular, we solve this problem in polynomial time for bipartite permutation graphs and threshold graphs. For both graph classes, the distortion can be arbitrarily large. The graphs that we consider are unweighted.

Finding Maximum Edge Bicliques in Convex Bipartite Graphs

A bipartite graph G=(A,B,E) is convex on B if there exists an ordering of the vertices of B such that for any vertex v∈A, vertices adjacent to v are consecutive in B. A complete bipartite subgraph of a graph G is called a biclique of G. Motivated by an application to analyzing DNA microarray data, we study the problem of finding maximum edge bicliques in convex bipartite graphs. Given a bipartite graph G=(A,B,E) which is convex on B, we present a new algorithm that computes a maximum edge biclique of G in O(nlog 3 nlog log n) time and O(n) space, where n=|A|. This improves the current O(n 2) time bound available for the problem. We also show that for two special subclasses of convex bipartite graphs, namely for biconvex graphs and bipartite permutation graphs, a maximum edge biclique can be computed in O(nα(n)) and O(n) time, respectively, where n=min (|A|,|B|) and α(n) is the slowly growing inverse of the Ackermann function.

The Recognition of Tolerance and Bounded Tolerance Graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This subclass of perfect graphs has been extensively studied, due to both its interesting structure and its numerous applications (in bioinformatics, constraint-based temporal reasoning, resource allocation, and scheduling problems, among others). Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance graphs. In spite of this, the recognition of tolerance graphs—namely, the problem of deciding whether a given graph is a tolerance graph—as well as the recognition of their main subclass of bounded tolerance graphs, have been the most fundamental open problems on this class of graphs (cf. the book on tolerance graphs [M. C. Golumbic and A. N. Trenk, Tolerance Graphs, Cambridge Stud. Adv. Math. 89, Cambridge University Press, Cambridge, UK, 2004]) since their introduction in 1982 [M. C. Golumbic and C. L. Monma, Proceedings of the 13th Southeastern Conference on Combinatorics, Graph Theory and Computing, Congr. Numer., 35 (1982), pp. 321–331]. In this article we prove that both recognition problems are NP-complete, even in the case where the input graph is a trapezoid graph. The presented results are surprising because, on the one hand, most subclasses of perfect graphs admit polynomial recognition algorithms and, on the other hand, bounded tolerance graphs were believed to be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time) and share a very similar structure with them. For our reduction we extend the notion of an acyclic orientation of permutation and trapezoid graphs. Our main tool is a new algorithm that uses vertex splitting to transform a given trapezoid graph into a permutation graph, while preserving this new acyclic orientation property. This method of vertex splitting is of independent interest; very recently, it was also proved a powerful tool in the design of efficient recognition algorithms for other classes of graphs [G. B. Mertzios and D. G. Corneil, Discrete Appl. Math., 159 (2011), pp. 1131–1147].

Functigraphs: An extension of permutation graphs

Let G1 and G2 be copies of a graph G, and let f : V (G1) ! V (G2) be a function. Then a functigraph C(G,f) = (V,E) is a generalization of a permutation graph, where V = V (G1) ( V (G2) and E = E(G1) ( E(G2) ( {uv: u 2 V (G1),v 2 V (G2), v = f(u)}. In this paper, we study colorability and planarity of functigraphs.

Canonical Antichains of Unit Interval and Bipartite Permutation Graphs

We study unit interval graphs and bipartite permutation graphs partially ordered by the induced subgraph relation. It is known that neither of these classes is well-quasi-ordered, since each of them contains an infinite antichain. We show that in both cases the antichains are canonical in the sense that any subclass of unit interval or bipartite permutation graphs containing only finitely many graphs from the respective antichain is well-quasi-ordered.

On Inverse Chromatic Number problems (Extended abstract)

Abstract We study inverse chromatic number problems in permutation graphs and in interval graphs. Given a fixed instance and a fixed integer K, the instance has to be modified as little as possible so that the newly obtained graph can be colored with K colors or less. We show that the inverse (p, k)-colorability problem (defined similarly) in permutation graphs is polynomially solvable for fixed p and k. We then propose a polynomial-time algorithm for solving inverse chromatic number problem in interval graphs where all intervals have length 1 or 2. We also show that the latter problem is NP-hard if there is a constant number of different interval lengths.

Treewidth and minimum fill-in on permutation graphs in linear time

Permutation graphs form a well-studied subclass of cocomparability graphs. Permutation graphs are the cocomparability graphs whose complements are also cocomparability graphs. A triangulation of a graph G is a graph H that is obtained by adding edges to G to make it chordal. If no triangulation of G is a proper subgraph of H then H is called a minimal triangulation. The main theoretical result of the paper is a characterisation of the minimal triangulations of a permutation graph, that also leads to a succinct and linear-time computable representation of the set of minimal triangulations. We apply this representation to devise linear-time algorithms for various minimal triangulation problems on permutation graphs, in particular, we give linear-time algorithms for computing treewidth and minimum fill-in on permutation graphs.

Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs

The densest k-subgraph problem asks for a k-vertex subgraph with the maximum number of edges. This problem is NP-hard on bipartite graphs, chordal graphs, and planar graphs. A 3-approximation algorithm is known for chordal graphs. We present 3 2 -approximation algorithms for proper interval graphs and bipartite permutation graphs. The latter result relies on a new characterisation of bipartite permutation graphs which may be of independent interest.

A New Intersection Model and Improved Algorithms for Tolerance Graphs

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs, which generalizes in a natural way both interval and permutation graphs, has attracted many research efforts since their introduction in [M. C. Golumbic and C. L. Monma, Congr. Numer., 35 (1982), pp. 321–331], as it finds many important applications in constraint-based temporal reasoning, resource allocation, and scheduling problems, among others. In this article we propose the first non-trivial intersection model for general tolerance graphs, given by three-dimensional parallelepipeds, which extends the widely known intersection model of parallelograms in the plane that characterizes the class of bounded tolerance graphs. Apart from being important on its own, this new representation also enables us to improve the time complexity of three problems on tolerance graphs. Namely, we present optimal $\mathcal{O}(n\log n)$ algorithms for computing a minimum coloring and a maximum clique and an $\mathcal{O}(n^{2})$ algorithm for computing a maximum weight independent set in a tolerance graph with n vertices, thus improving the best known running times $\mathcal{O}(n^{2})$ and $\mathcal{O}(n^{3})$ for these problems, respectively.

Additive tree 2-spanners of permutation graphs

Let G be a connected graph. A spanning tree T of G is a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. If their distances in T and G differ by at most t, then T is an additive tree t-spanner of G. In this paper, we show that any permutation graph has an additive tree 2-spanner, and it can be found in O(n) time sequentially or in O(log n) time with O(n/log n) processors on the EREW PRAM computational model by using a previously published algorithm for finding a tree 3-spanner of a permutation graph.

BOUNDS ON THE DOMINATION NUMBER OF PERMUTATION GRAPHS

For a graph G with n vertices and a permutation α on V(G), a permutation graph Pα(G) is obtained from two identical copies of G by adding an edge between v and α(V) for any v ϵ V(G). Let γ(G) be the domination number of a graph G. It has been shown that γ(G) ≤ γ(Pα(G) ≤ 2γ(G) for any permutation α on V(G). In this paper, we investigate specific graphs for which there exists a permutation α such that γ(Pα(G)) ≻ γ(G) in terms of the domination number of G or the maximum degree of G. Additionally, we construct a class of graphs for which the domination number of any permutation graph is twice the domination number of the original graph, as well as explore finding a specific graph G and permutation α for any two positive integers a and b with a ≤ b ≤ 2a, to have γ(G) = a and γ(Pα(G)) = b.

On the complexity of signed and minus total domination in graphs

In this paper we present unified methods to solve the minus and signed total domination problems for chordal bipartite graphs and trees in O ( n 2 ) and O ( n + m ) time, respectively. We also prove that the decision problem for the signed total domination problem on doubly chordal graphs is NP-complete. Note that bipartite permutation graphs, biconvex bipartite graphs, and convex bipartite graphs are subclasses of chordal bipartite graphs.

Labeling bipartite permutation graphs with a condition at distance two

An L ( p , q ) -labeling of a graph G is an assignment f from vertices of G to the set of non-negative integers { 0 , 1 , … , λ } such that | f ( u ) − f ( v ) | ≥ p if u and v are adjacent, and | f ( u ) − f ( v ) | ≥ q if u and v are at distance 2 apart. The minimum value of λ for which G has L ( p , q ) -labeling is denoted by λ p , q ( G ) . The L ( p , q ) -labeling problem is related to the channel assignment problem for wireless networks. In this paper, we present a polynomial time algorithm for computing L ( p , q ) -labeling of a bipartite permutation graph G such that the largest label is at most ( 2 p − 1 ) + q ( b c ( G ) − 2 ) , where b c ( G ) is the biclique number of G . Since λ p , q ( G ) ≥ p + q ( b c ( G ) − 2 ) for any bipartite graph G , the upper bound is at most p − 1 far from optimal.

Treelike comparability graphs

An undirected graph is a treelike comparability graph if it admits a transitive orientation such that its transitive reduction is a tree. We show that treelike comparability graphs are distance hereditary. Utilizing this property, we give a linear time recognition algorithm. We then characterize permutation graphs that are treelike. Finally, we consider the Partitioning into Bounded Cliques problem on special subgraphs of treelike permutation graphs.

On probe permutation graphs

Given a class of graphs G , a graph G is a probe graph of G if its vertices can be partitioned into two sets, P , the probes, and an independent set N , the nonprobes, such that G can be embedded into a graph of G by adding edges between certain vertices of N . If the partition of the vertices into probes and nonprobes is part of the input, then we call the graph a partitioned probe graph of G . In this paper, we provide a recognition algorithm for partitioned probe permutation graphs with time complexity O ( n 2 ) , where n is the number of vertices of the input graph. We show that a probe permutation graph has at most O ( n 4 ) minimal separators. As a consequence, for probe permutation graphs there exist polynomial-time algorithms solving problems like treewidth and minimum fill-in. We also characterize those graphs for which the probe graphs must be weakly chordal.