A linear time recognition algorithm for proper interval graphs

We propose a linear time recognition algorithm for proper interval graphs. The algorithm is based on certain ordering of vertices, called bicompatible elimination ordering (BCO). Given a BCO of a biconnected proper interval graph G, we also propose a linear time algorithm to construct a Hamiltonian cycle of G.     

Scale free interval graphs

Scale free graphs have attracted attention by their non-uniform structure that can be used as a model for various social and physical networks. In this paper, we propose a natural and simple random model for generating scale free interval graphs. The model generates a set of intervals randomly under a certain distribution, which defines a random interval graph. The main advantage of the model is its simpleness. The structure/properties of generated graphs are analyzable by relatively simple probabilistic and/or combinatorial arguments, which is different from many other models. Based on such arguments, we show for our random interval graph that its degree distribution follows a power law, and that it has a large average clustering coefficient.     

Editing to Eulerian graphs

The Eulerian Editing problem asks, given a graph G and an integer k, whether G can be modified into an Eulerian graph using at most k edge additions and edge deletions. We show that this problem is polynomial-time solvable for both undirected and directed graphs. We generalize these results for problems with degree parity constraints and degree balance constraints, respectively. We also consider the variants where vertex deletions are permitted. Combined with known results, this leads to full complexity classifications for both undirected and directed graphs and for every subset of the three graph operations.     

Finding hidden independent sets in interval graphs

We design efficient competitive algorithms for discovering hidden information using few queries. Specifically, consider a game in a given set of intervals (and their implied interval graph G) in which our goal is to discover an (unknown) independent set X by making the fewest queries of the form “Is point p covered by an interval in X?” Our interest in this problem stems from two applications: experimental gene discovery with PCR technology and the game of Battleship (in a 1-dimensional setting). We provide adaptive algorithms for both the verification scenario (given an independent set, is it X?) and the discovery scenario (find X without any information). Under some assumptions, these algorithms use an asymptotically optimal number of queries in every instance.     

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.     

An optimal algorithm for shortest paths on weighted interval and circular-arc graphs,with applications

We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-arc graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcsand the sorted list of the interval endpoints. Our algorithm solves this problem optimally inO(n) time, wheren is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is anO(qn + n logn)-time algorithm for the minimum-weight circle-cover problem, whereq is the minimum number of arcs crossing any point on the circle; then logn term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were0(n logn) for this shortest paths problem, andO(qn logn) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar (logn)-factor improvements in other problems on such graphs.The research of M. J. Atallah was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the Air Force Office of Scientific Research under Contract AFOSR-90-0107, and by the National Science Foundation under Grant CCR-9202807. D. Z. Chen's research was supported in part by the Leonardo Fibonacci Institute, Trento, Italy. The research of D. T. Lee was supported in part by the Leonardo Fibonacci Institute, Trento, Italy, by the National Science Foundation, and the Office of Naval Research under Grants CCR-8901815, CCR-9309743, and N00014-93-1-0272.     

Packing triangles in bounded degree graphs

We consider the two problems of finding the maximum number of node disjoint triangles and edge disjoint triangles in an undirected graph. We show that the first (respectively second) problem is polynomially solvable if the maximum degree of the input graph is at most 3 (respectively 4), whereas it is APX-hard for general graphs and NP-hard for planar graphs if the maximum degree is 4 (respectively 5) or more.     

Efficient parallel recognition of some circular arc graphs,I

We present the first efficient parallel algorithms for recognizing some subclasses of circular arc graphs including circular arc graphs and proper interval graphs. These algorithms run in O(log² n) time withO(n ³) processors on a CRCW PRAM. An intersection representation can also be constructed within the same resource bounds. Furthermore, we propose some new characterizations of circular arc graphs and proper interval graphs.Portions of this paper have appeared in preliminary form in theProceedings of the 1989 IEEE international Symposium on Circuits and Systems [9], theProceedings of the 1989 Workshop on Algorithms and Data Structures [10], and theProceedings of the 1990 Canadian Conference on Computational Geometry [11].     

The clique-transversal set problem in claw-free graphs with degree at most 4

A clique of a graph G is defined as a complete subgraph maximal under inclusion and having at least two vertices. A clique-transversal set D of G is a subset of vertices of G such that D meets all cliques of G. The clique-transversal set problem is to find a minimum clique-transversal set of G. In this paper we present a polynomial time algorithm for the clique-transversal set problem on claw-free graphs with degree at most 4.     

Degree-constrained decompositions of graphs: Bounded treewidth and planarity

We study the problem of decomposing the vertex set V$V$ of a graph into two nonempty parts _V1,_V2$V_1,V_2$ which induce subgraphs where each vertex v∈_V1$v\in V_1$ has degree at least a(v)$a(v)$ inside _V1$V_1$ and each v∈_V2$v\in V_2$ has degree at least b(v)$b(v)$ inside _V2$V_2$. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This result and its variants are then applied to designing polynomial-time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible.     

Approximating maximum weight K-colorable subgraphs in chordal graphs

We present a 2-approximation algorithm for the problem of finding the maximum weight K-colorable subgraph in a given chordal graph with node weights. The running time of the algorithm is O(K(n+m)), where n and m are the number of vertices and edges in the given graph.     

An optimal parallel algorithm for computing cut vertices and blocks on interval graphs

In this paper, a parallel algorithm is presented to find all cut-vertices and blocks of an interval graph. If the list of sorted end points of the intervals of an interval graph is given then the proposed algorithm takes O(log n) time and O(n/log n) processors on an EREW PRAM, if the sorted list is not given then the time and processors complexities are respectively O(log n) and O(n).     

Honeycomb toroidal graphs are Cayley graphs

There is a particular family of trivalent vertex-transitive graphs that have been called generalized honeycomb tori by some and brick products by others. They have been studied as hexagonal embeddings on the torus as well. We show that all these graphs are Cayley graphs on generalized dihedral groups.     

Graphs of interval count two with a given partition

We describe a polynomial time algorithm to decide for a given connected graph G and a given partition of its vertex set into two sets A and B  , whether it is possible to assign a closed interval I(u)$I(u)$ to each vertex u of G such that two distinct vertices u and v of G   are adjacent if and only if I(u)$I(u)$ and I(v)$I(v)$ intersect, all intervals assigned to vertices in A   have some length _LA$L_A$, and all intervals assigned to vertices in B   have some length _LB$L_B$ where _LA<_LB$L_A<L_B$. Our result is motivated by the interval count problem whose complexity status is open.     

A linear-time algorithm to compute a MAD tree of an interval graph

The average distance of a connected graph G is the average of the distances between all pairs of vertices of G. We present a linear time algorithm that determines, for a given interval graph G, a spanning tree of G with minimum average distance (MAD tree). Such a tree is sometimes referred to as a minimum routing cost spanning tree.     

Fault diameter of product graphs

The (k−1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of an induced subgraph by deleting at most k−1 vertices from G. This paper considers the fault diameter of the product graph G₁∗G₂ of two graphs G₁ and G₂ and proves that Dk₁+k₂(G₁∗G₂)?Dk₁(G₁)+Dk₂(G₂)+1 if G₁ is k₁-connected and G₂ is k₂-connected. This generalizes some known results such as Bani? and ?erovnik [I. Bani?, J. ?erovnik, Fault-diameter of Cartesian graph bundles, Inform. Process. Lett. 100 (2) (2006) 47-51].