On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity “within a given type.”     

Eulerian detachments with local edge-connectivity

For a graph G, a detachment operation at a vertex transforms the graph into a new graph by splitting the vertex into several vertices in such a way that the original graph can be obtained by contracting all the split vertices into a single vertex. A graph obtained from a given graph G by applying detachment operations at several vertices is called a detachment of graph G. While detachment operations may decrease the connectivity of graphs, there are several works on conditions for preserving the connectivity. In this paper, we present necessary and sufficient conditions for a given graph/digraph to have an Eulerian detachment that satisfies a given local edge-connectivity requirement. We also discuss conditions for the detachment to be loopless.     

Clique-critical graphs: Maximum size and recognition

The clique graph of G, K(G), is the intersection graph of the family of cliques (maximal complete sets) of G. Clique-critical graphs were defined as those whose clique graph changes whenever a vertex is removed. We prove that if G has m edges then any clique-critical graph in K^-1(G) has at most 2m vertices, which solves a question posed by Escalante and Toft [On clique-critical graphs, J. Combin. Theory B 17 (1974) 170-182]. The proof is based on a restatement of their characterization of clique-critical graphs. Moreover, the bound is sharp. We also show that the problem of recognizing clique-critical graphs is NP-complete.     

Coalescent random walks on graphs

Inspired by coalescent theory in biology, we introduce a stochastic model called “multi-person simple random walks” or “coalescent random walks” on a graph G. There are any finite number of persons distributed randomly at the vertices of G. In each step of this discrete time Markov chain, we randomly pick up a person and move it to a random adjacent vertex. To study this model, we introduce the tensor powers of graphs and the tensor products of Markov processes. Then the coalescent random walk on graph G becomes the simple random walk on a tensor power of G. We give estimates of expected number of steps for these persons to meet all together at a specific vertex. For regular graphs, our estimates are exact.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

Vertex pancyclicity in quasi-claw-free graphs

A graph G is called quasi-claw-free if for any two vertices x and y with distance two there exists a vertex u∈N(x)∩N(y) such that N[u]⊆N[x]∪N[y]. This concept is a natural extension of the classical claw-free graphs. In this paper, we present two sufficient conditions for vertex pancyclicity in quasi-claw-free graphs, namely, quasilocally connected and almost locally connected graphs. Our results include some well-known results on claw-free graphs as special cases. We also give an affirmative answer to a problem proposed by Ainouche.     

Some Remarks on Rainbow Connectivity

An edge (vertex) colored graph is rainbow‐connected if there is a rainbow path between any two vertices, i.e. a path all of whose edges (internal vertices) carry distinct colors. Rainbow edge (vertex) connectivity of a graph G is the smallest number of colors needed for a rainbow edge (vertex) coloring of G. In this article, we propose a very simple approach to studying rainbow connectivity in graphs. Using this idea, we give a unified proof of several known results, as well as some new ones.     

An extension of regular supermagic graphs

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we consider an extension of regular supermagic graphs and apply it to some constructions of supermagic graphs. Using the extension we prove that for any graph G there is a supermagic regular graph which contains an induced subgraph isomorphic to G.     

Total domination dot-stable graphs

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.     

The degree of a typical vertex in generalized random intersection graph models

In this paper we consider the degree of a typical vertex in two models of random intersection graphs introduced in [E. Godehardt, J. Jaworski, Two models of random intersection graphs for classification, in: M. Schwaiger, O. Opitz (Eds.), Exploratory Data Analysis in Empirical Research, Proceedings of the 25th Annual Conference of the Gesellschaft für Klassifikation e.V., University of Munich, March 14-16, 2001, Springer, Berlin, Heidelberg, New York, 2002, pp. 67-81], the active and passive models. The active models are those for which vertices are assigned a random subset of a list of objects and two vertices are made adjacent when their subsets intersect. We prove sufficient conditions for vertex degree to be asymptotically Poisson as well as closely related necessary conditions. We also consider the passive model of intersection graphs, in which objects are vertices and two objects are made adjacent if there is at least one vertex in the corresponding active model “containing” both objects. We prove a necessary condition for vertex degree to be asymptotically Poisson for passive intersection graphs.     

Every minor-closed property of sparse graphs is testable

Suppose G is a graph of bounded degree d, and one needs to remove ?n of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G^′. In fact, a similar result is proved for any minor-closed property of bounded degree graphs.The main motivation of the above result comes from theoretical computer-science. Using our main result we infer that for any minor-closed property P, there is a constant time algorithm for detecting if a graph is “far” from satisfying P. This, in particular, answers an open problem of Goldreich and Ron [STOC 1997] [20], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.     

Upper and lower bounds for the energy of bipartite graphs

Using Lagrange's multiplier rule, we find upper and lower bounds of the energy of a bipartite graph G, in terms of the number of vertices, edges and the spectral moment of fourth order. Moreover, the upper bound is attained in a graph G if and only if G is the graph of a symmetric balanced incomplete block design (BIBD). Also, we determine the graphs for which the lower bound is sharp.     

Subspace intersection graphs

Given a set R of affine subspaces in R^d of dimension e, its intersection graph G has a vertex for each subspace, and two vertices are adjacent in G if and only if their corresponding subspaces intersect. For each pair of positive integers d and e we obtain the class of (d,e)-subspace intersection graphs. We classify the classes of (d,e)-subspace intersection graphs by containment, for e=1 or e=d−1 or d≤4.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G=(V,E) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G. The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O(|V|⁹) time, to solve the path cover problem on distance-hereditary graphs.     

Whiskers and sequentially Cohen-Macaulay graphs

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen-Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G∪W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen-Macaulay graph. For instance, we show that if S is a vertex cover of G, then G∪W(S) is a sequentially Cohen-Macaulay graph. On the other hand, we show that if G?S is not sequentially Cohen-Macaulay, then G∪W(S) is not a sequentially Cohen-Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen-Macaulay graphs.     

Tree 3-spanners in 2-sep directed path graphs: Characterization, recognition, and construction

A spanning tree T of a graph G is called a treet-spanner, if the distance between any two vertices in T is at most t-times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4, and is linearly solvable for t=1 and t=2. The case t=3 still remains open. A directed path graph is called a 2-sep directed path graph if all of its minimal a−b vertex separator for every pair of non-adjacent vertices a and b are of size two. Le and Le [H.-O. Le, V.B. Le, Optimal tree 3-spanners in directed path graphs, Networks 34 (2) (1999) 81-87] showed that directed path graphs admit tree 3-spanners. However, this result has been shown to be incorrect by Panda and Das [B.S. Panda, Anita Das, On tree 3-spanners in directed path graphs, Networks 50 (3) (2007) 203-210]. In fact, this paper observes that even the class of 2-sep directed path graphs, which is a proper subclass of directed path graphs, need not admit tree 3-spanners in general. It, then, presents a structural characterization of tree 3-spanner admissible 2-sep directed path graphs. Based on this characterization, a linear time recognition algorithm for tree 3-spanner admissible 2-sep directed path graphs is presented. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep directed path graph is proposed.     

Representing edge intersection graphs of paths on degree 4 trees

Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P, denoted by EPT(P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G=EPT(P) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph.An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.     

Every 3‐connected claw‐free Z8‐free graph is Hamiltonian

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. Given a graph G, the biclique matrix of G is a {0,1,?1} matrix having one row for each biclique and one column for each vertex of G, and such that a pair of 1, ?1 entries in a same row corresponds exactly to adjacent vertices in the corresponding biclique. We describe a characterization of biclique matrices, in similar terms as those employed in Gilmore's characterization of clique matrices. On the other hand, the biclique graph of a graph is the intersection graph of the bicliques of G. Using the concept of biclique matrices, we describe a Krausz‐type characterization of biclique graphs. Finally, we show that every induced P₃ of a biclique graph must be included in a diamond or in a 3‐fan and we also characterize biclique graphs of bipartite graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 1–16, 2010