Proper Interval Graphs Research Articles

Rooted directed path graphs are leaf powers

Leaf powers are a graph class which has been introduced to model the problem of reconstructing phylogenetic trees. A graph G = ( V , E ) is called k -leaf power if it admits a k -leaf root, i.e., a tree T with leaves V such that u v is an edge in G if and only if the distance between u and v in T is at most k . Moroever, a graph is simply called leaf power if it is a k -leaf power for some k ∈ N . This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a given graph is a k -leaf power. We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a well-known graph class with absolutely different motivations. After this, we provide the largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path graphs. Subsequently, we study the leaf rank problem, the algorithmic challenge of determining the minimum k for which a given graph is a k -leaf power. Firstly, we give a lower bound on the leaf rank of a graph in terms of the complexity of its separators. Secondly, we use this measure to show that the leaf rank is unbounded on both the class of ptolemaic and the class of unit interval graphs. Finally, we provide efficient algorithms to compute 2 | V | -leaf roots for given ptolemaic or (unit) interval graphs G = ( V , E ) .

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D . The geodetic number of G is the minimum cardinality of a geodetic set of G . We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k . Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.

ON DISTANCE TWO LABELLING OF UNIT INTERVAL GRAPHS

An $L(2,1)$-labelling of a graph $G$ is an assignment of non-negative integers to the vertices of $G$ such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The $L(2,1)$-labelling number of $G$, denoted by $\lambda(G)$, is the minimum range of labels over all such labellings. In this paper, we first discuss some necessary and sufficient conditions for unit interval graph $G$ to have $\lambda(G)=2\chi(G)-2$ and then characterize all unit interval graphs $G$ of order no more than $3\chi(G)-1$, where $\chi(G)$ is the chromatic number of $G$. Finally, we discuss some subgraphs of unit interval graphs $G$ on more than $2\chi(G)+1$ vertices with $\lambda(G) = 2\chi(G)$.

A simple algorithm to find Hamiltonian cycles in proper interval graphs

We present an algorithm to find a Hamiltonian cycle in a proper interval graph in O ( m + n ) time, where m is the number of edges and n is the number of vertices in the graph. The algorithm is simpler and shorter than previous algorithms for the problem.

A parallel algorithm for generating bicompatible elimination orderings of proper interval graphs

In this paper, we first show how a certain ordering of vertices, called bicompatible elimination ordering (BCO), of a proper interval graph can be used to solve efficiently several problems in proper interval graphs. We, then, propose an NC parallel algorithm (i.e., polylogarithmic-time employing a polynomial number of processors) in SIMD CRCW PRAM (Single Instruction Stream Multiple Data Stream Concurrent Read Concurrent Write Parallel Random Access Machine) model of parallel computation to compute a BCO of a proper interval graph. To the best of our knowledge, this is the first NC parallel algorithm to compute a BCO of a proper interval graph.

Improper coloring of unit disk graphs

AbstractMotivated by a satellite communications problem, we consider a generalized coloring problem on unit disk graphs. A coloring is k‐improper if no more than k neighbors of every vertex have the same colour as that assigned to the vertex. The k‐improper chromatic number χk(G) is the least number of colors needed in a k‐improper coloring of a graph G. The main subject of this work is analyzing the complexity of computing χk for the class of unit disk graphs and some related classes, e.g., hexagonal graphs and interval graphs. We show NP‐completeness in many restricted cases and also provide both positive and negative approximability results. Because of the challenging nature of this topic, many seemingly simple questions remain: for example, it remains open to determine the complexity of computing χk for unit interval graphs. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

The clique-separator graph for chordal graphs

We present a new representation of a chordal graph called the clique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph. We also characterize proper interval graphs and split graphs in terms of the clique-separator graph. We present an algorithm that constructs the clique-separator graph of a chordal graph in O ( n 3 ) time and of an interval graph in O ( n 2 ) time, where n is the number of vertices in the graph.

A new representation of proper interval graphs with an application to clique-width

Abstract We introduce a new representation of proper interval graphs that can be computed in linear time and stored in O ( n ) space. This representation is a 2-dimensional vertex partition. It is particularly interesting with respect to clique-width. Based on this representation, we prove new upper bounds on the clique-width of proper interval graphs.

FAST EXPONENTIAL-TIME ALGORITHMS FOR THE FOREST COUNTING AND THE TUTTE POLYNOMIAL COMPUTATION IN GRAPH CLASSES

We prove # P -completeness for counting the number of forests in regular graphs and chordal graphs. We also present algorithms for this problem, running in O *(1.8494m) time for 3-regular graphs, and O *(1.9706m) time for unit interval graphs, where m is the number of edges in the graph and O *-notation ignores a polynomial factor. The algorithms can be generalized to the Tutte polynomial computation.

Minimum vertex ranking spanning tree problem for chordal and proper interval graphs

A vertex k-ranking of a simple graph is a coloring of its ver- tices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et. al. proved in (Np-hardness proof and an approximation algorithm for the minimum vertex ranking span- ning tree problem, Discrete Appl. Math. 154 (2006) 2402-2410) that the decision problem: given a simple graph G, decide whether there exists a spanning tree T of G such that T has a vertex 4-ranking, is NP- complete. In this paper we improve this result by proving NP-hardness of finding for a given chordal graph its spanning tree having v ertex 3- ranking. This bound is the best possible. On the other hand we prove that MVRST problem can be solved in linear time for proper interval graphs.

A dynamic distributed approach to representing proper interval graphs

First studied by Brodal and Fagerberg [G.S. Brodal, R. Fagerberg, Dynamic representation of sparse graphs, in: Algorithms and Data Structures, Proceedings of the 6th International Workshop, Vancouver, Canada, in: Lecture Notes in Computer Science, vol. 1663, Springer-Verlag, 1999], a dynamic adjacency labelling scheme labels the vertices of a graph so that the adjacency of two vertices can be deduced from their labels. The scheme is dynamic in the sense that only a small adjustment must be made to the vertex labels when a small change is made to the graph. Using a centralized dynamic representation of Hell, Shamir and Sharan [P. Hell, R. Shamir, R. Sharan, A fully dynamic algorithm for recognizing and representing proper interval graphs, SIAM Journal on Computing 31 (1) (2001) 289–305], we develop a O ( log n ) bit/label dynamic adjacency labelling scheme for proper interval graphs. Our fully dynamic scheme handles vertex deletion/addition and edge deletion/addition in O ( n ) time. Furthermore, our dynamic scheme is error-detecting, as it recognizes when the new graph is not a proper interval graph.

On the computation of the hull number of a graph

Let G be a graph. If u , v ∈ V ( G ) , a u – v shortest path of G is a path linking u and v with minimum number of edges. The closed interval I [ u , v ] consists of all vertices lying in some u – v shortest path of G . For S ⊆ V ( G ) , the set I [ S ] is the union of all sets I [ u , v ] for u , v ∈ S . We say that S is a convex set if I [ S ] = S . The convex hull of S , denoted I h [ S ] , is the smallest convex set containing S . A set S is a hull set of G if I h [ S ] = V ( G ) . The cardinality of a minimum hull set of G is the hull number of G , denoted by h n ( G ) . In this work we prove that deciding whether h n ( G ) ≤ k is NP-complete.We also present polynomial-time algorithms for computing h n ( G ) when G is a unit interval graph, a cograph or a split graph.

Representation of graphs by OBDDs

Recently, it has been shown in a series of works that the representation of graphs by Ordered Binary Decision Diagrams (OBDDs) often leads to good algorithmic behavior. However, the question for which graph classes an OBDD representation is advantageous, has not been investigated, yet. In this paper, the space requirements for the OBDD representation of certain graph classes, specifically cographs, several types of graphs with few P 4 s, unit interval graphs, interval graphs and bipartite graphs are investigated. Upper and lower bounds are proven for all these graph classes and it is shown that in most (but not all) cases a representation of the graphs by OBDDs is advantageous with respect to space requirements.

A dichotomy for minimum cost graph homomorphisms

For graphs G and H , a mapping f : V ( G ) → V ( H ) is a homomorphism of G to H if u v ∈ E ( G ) implies f ( u ) f ( v ) ∈ E ( H ) . If, moreover, each vertex u ∈ V ( G ) is associated with costs c i ( u ) , i ∈ V ( H ) , then the cost of the homomorphism f is ∑ u ∈ V ( G ) c f ( u ) ( u ) . For each fixed graph H , we have the minimum cost homomorphism problem , written as MinHOM ( H ) . The problem is to decide, for an input graph G with costs c i ( u ) , u ∈ V ( G ) , i ∈ V ( H ) , whether there exists a homomorphism of G to H and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well-studied optimization problems. We prove a dichotomy of the minimum cost homomorphism problems for graphs H , with loops allowed. When each connected component of H is either a reflexive proper interval graph or an irreflexive proper interval bigraph, the problem MinHOM ( H ) is polynomial time solvable. In all other cases the problem MinHOM ( H ) is NP-hard. This solves an open problem from an earlier paper.

Minimal proper interval completions

Given an arbitrary graph G = ( V , E ) and a proper interval graph H = ( V , F ) with E ⊆ F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any sandwich graph H ′ = ( V , F ′ ) with E ⊆ F ′ ⊂ F , H ′ is not a proper interval graph. In this paper we give a O ( n + m ) time algorithm computing a minimal proper interval completion of an arbitrary graph. The output is a proper interval model of the completion.

Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles ( VTP ) and edge-disjoint triangles ( ETP ) in a simple graph. Both problems are NP -hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3 / 2 + ɛ , a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68–72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX -hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

A matrix characterization of interval and proper interval graphs

In this work a matrix representation that characterizes the interval and proper interval graphs is presented, which is useful for the efficient formulation and solution of optimization problems, such as the k -cluster problem. For the construction of this matrix representation every such graph is associated with a node versus node zero–one matrix. In contrast to representations used in most of the previous work, the proposed matrix characterization does not make use of the maximal cliques in the graph investigated.

Linear time algorithms for finding sparsest cuts in various graph classes

Abstract [ S , S ¯ ] denotes the set of edges with exactly one end vertex in S. The density of an edge cut [ S , S ¯ ] is | [ S , S ¯ ] | / ( | S | | S ¯ | ) . A sparsest cut is an edge cut with minimum density. We characterize the sparsest cuts for unit interval graphs, complete bipartite graphs and cactus graphs. For all of these classes, the characterizations lead to linear time algorithms to find a sparsest cut.

The Roberts characterization of proper and unit interval graphs

In this note, a constructive proof that the classes of proper interval graphs and unit interval graphs coincide is given, a result originally established by Fred S. Roberts. Additionally, the proof yields a linear-time and space algorithm to compute a unit interval representation, given a proper interval graph as input.

Approximate L(δ1,δ2,…,δt)‐coloring of trees and interval graphs

AbstractGiven a vector (δ1,δ2,…,δt) of nonincreasing positive integers, and an undirected graph G = (V,E), an L(δ1,δ2,…,δt)‐coloring of G is a function f from the vertex set V to a set of nonnegative integers such that ∣f(u) − f(v)∣ ≥ δi, if d(u,v) = i, 1 ≤ i ≤ t, where d(u,v) is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal L(δ1,δ2,…,δt)‐coloring for G is one minimizing the largest integer used over all such colorings. Such a coloring problem has relevant applications in channel assignment for interference avoidance in wireless networks. This article presents efficient approximation algorithms for L(δ1,δ2,…,δt)‐coloring of two relevant classes of graphs—trees, and interval graphs. Specifically, based on the notion of strongly simplicial vertices, O(n(t + δ1)) and O(nt2δ1) time algorithms are proposed to find α‐approximate colorings on interval graphs and trees, respectively, where n is the number of vertices and α is a constant depending on t and δ1,…,δt. Moreover, an O(n) time algorithm is given for the L(δ1,δ2)‐coloring of unit interval graphs, which provides a 3‐approximation. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 49(3), 204–216 2007