The d-precoloring problem for k-degenerate graphs

In this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXT problem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial k-trees (without any size restriction on S). We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees.     

splitting number is NP-complete

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k⩾0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v₁ and v₂, and attaches the neighbors of v either to v₁ or to v₂. We prove that the splitting number decision problem is NP-complete when restricted to cubic graphs. We obtain as a consequence that planar subgraph remains NP-complete when restricted to cubic graphs. Note that NP-completeness for cubic graphs implies NP-completeness for graphs not containing a subdivision of K₅ as a subgraph.     

Clustering and domination in perfect graphs

A k-cluster in a graph is an induced subgraph on k vertices which maximizes the number of edges. Both the k-cluster problem and the k-dominating set problem are NP-complete for graphs in general. In this paper we investigate the complexity status of these problems on various sub-classes of perfect graphs. In particular, we examine comparability graphs, chordal graphs, bipartite graphs, split graphs, cographs and κ-trees. For example, it is shown that the k-cluster problem is NP-complete for both bipartite and chordal graphs and the independent k-dominating set problem is NP-complete for bipartite graphs. Furthermore, where the k-cluster problem is polynomial we study the weighted and connected versions as well. Similarly we also look at the minimum k-dominating set problem on families which have polynomial k-dominating set algorithms.     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.     

An extension of Sallee’s theorem to infinite locally finite vap-free plane graphs

A graph isk-cyclable if givenk vertices there is a cycle that contains thek vertices. Sallee showed that every finite 3-connected planar graph is 5-cyclable. In this paper, by characterizing the circuit graphs and investigating the structure of LV-graphs, we extend his result to 3-connected infinite locally finite VAP-free plane graphs.     

Large non-planar graphs and an application to crossing-critical graphs

We prove that, for every positive integer k, there is an integer N such that every 4-connected non-planar graph with at least N vertices has a minor isomorphic to K4,k, the graph obtained from a cycle of length 2k+1 by adding an edge joining every pair of vertices at distance exactly k, or the graph obtained from a cycle of length k by adding two vertices adjacent to each other and to every vertex on the cycle. We also prove a version of this for subdivisions rather than minors, and relax the connectivity to allow 3-cuts with one side planar and of bounded size. We deduce that for every integer k there are only finitely many 3-connected 2-crossing-critical graphs with no subdivision isomorphic to the graph obtained from a cycle of length 2k by joining all pairs of diagonally opposite vertices.     

The parameterized complexity of the induced matching problem

Given a graph G and an integer k≥0, the NP-complete Induced Matching problem asks whether there exists an edge subset M of size at least k such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs, as well as on many restricted graph classes has been studied intensively. However, other than the fact that the problem is W[1]-hard on general graphs, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we provide first-time fixed-parameter tractability results for planar graphs, bounded-degree graphs, graphs with girth at least six, bipartite graphs, line graphs, and graphs of bounded treewidth. In particular, we give a linear-size problem kernel for planar graphs.     

Anticoloring and separation of graphs

An anticoloring of a graph is a coloring of some of the vertices, such that no two adjacent vertices are colored in distinct colors. The anticoloring problem seeks, roughly speaking, such colorings with many vertices colored in each color. We deal with the anticoloring problem for planar graphs and, using Lipton and Tarjan’s separation algorithm, provide an algorithm with some bound on the error. We also show that, to solve the anticoloring problem for general graphs, it suffices to solve it for connected graphs.     

A Turán-Type Problem on Distances in Graphs

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices pairwise at distance k has at most (n ? k + 1)²/4 pairs of vertices at distance k.     

Intersection graphs of curves in the plane

Let V be a set of curves in the plane. The corresponding intersection graph has V as the set of vertices, and two vertices are connected by an edge if and only if the two corresponding curves intersect in the plane.It is shown that the set of intersection graphs of curves in the plane is a proper subset of the set of all undirected graphs. Furthermore, the set of intersection graphs of straight line-segments is a proper subset of the set of the intersection graphs of curves in the plane. Finally, it is shown that for every k ≥ 3, the problem of determining whether an intersection graph of straight line-segments is k-colorable is NP-complete.     

Network flow interdiction on planar graphs

The network flow interdiction problem asks to reduce the value of a maximum flow in a given network as much as possible by removing arcs and vertices of the network constrained to a fixed budget. Although the network flow interdiction problem is strongly NP-complete on general networks, pseudo-polynomial algorithms were found for planar networks with a single source and a single sink and without the possibility to remove vertices. In this work, we introduce pseudo-polynomial algorithms that overcome various restrictions of previous methods. In particular, we propose a planarity-preserving transformation that enables incorporation of vertex removals and vertex capacities in pseudo-polynomial interdiction algorithms for planar graphs. Additionally, a new approach is introduced that allows us to determine in pseudo-polynomial time the minimum interdiction budget needed to remove arcs and vertices of a given network such that the demands of the sink node cannot be completely satisfied anymore. The algorithm works on planar networks with multiple sources and sinks satisfying that the sum of the supplies at the sources equals the sum of the demands at the sinks. A simple extension of the proposed method allows us to broaden its applicability to solve network flow interdiction problems on planar networks with a single source and sink having no restrictions on the demand and supply. The proposed method can therefore solve a wider class of flow interdiction problems in pseudo-polynomial time than previous pseudo-polynomial algorithms and is the first pseudo-polynomial algorithm that can solve non-trivial planar flow interdiction problems with multiple sources and sinks. Furthermore, we show that the k-densest subgraph problem on planar graphs can be reduced to a network flow interdiction problem on a planar graph with multiple sources and sinks and polynomially bounded input numbers.     

Tree components in random distance graphs of special form

In this work, we consider the problem of distribution of the number of tree components with given number of vertices k(k ?? 2) for a certain series of random distance graphs. Generalizations of the classical Erd?s?CRényi results are obtained in the case of geometric graphs of special form.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

Four gravity results

The gravity of a graph H in a given family of graphs H is the greatest integer n with the property that for every integer m, there exists a supergraph G∈H of H such that each subgraph of G, which is isomorphic to H, contains at least n vertices of degree ?m in G. Madaras and Škrekovski introduced this concept and showed that the gravity of the path P_k on k?2 vertices in the family of planar graphs of minimum degree 2 is k-2 for each k≠5,7,8,9. They conjectured that for each of the four excluded cases the gravity is k-3. In this paper we show that this holds.     

Degree distribution in random planar graphs

We prove that for each k?0, the probability that a root vertex in a random planar graph has degree k tends to a computable constant dk, so that the expected number of vertices of degree k is asymptotically dkn, and moreover that k_∑dk=1. The proof uses the tools developed by Giménez and Noy in their solution to the problem of the asymptotic enumeration of planar graphs, and is based on a detailed analysis of the generating functions involved in counting planar graphs. However, in order to keep track of the degree of the root, new technical difficulties arise. We obtain explicit, although quite involved expressions, for the coefficients in the singular expansions of the generating functions of interest, which allow us to use transfer theorems in order to get an explicit expression for the probability generating function p(w)=k_∑dkwk. From this we can compute the dk to any degree of accuracy, and derive the asymptotic estimate dk∼c⋅k−1/2qk for large values of k, where q≈0.67 is a constant defined analytically.     

The Number of Labeled Outerplanar <Emphasis Type="Italic">k</Emphasis>-Cyclic Graphs

A k-cyclic graph is a graph with cyclomatic number k. An explicit formula for the number of labeled connected outerplanar k-cyclic graphs with a given number of vertices is obtained. In addition, such graphs with fixed cyclomatic number k and a large number of vertices are asymptotically enumerated. As a consequence, it is found that, for fixed k, almost all labeled connected outerplanar k-cyclic graphs with a large number of vertices are cacti.     

Identities for non-crossing graphs and multigraphs

We refine an identity between the numbers of certain non-crossing graphs and multigraphs, by modifying a bijection found by P. Podbrdský [A bijective proof of an identity for noncrossing graphs, Discrete Math. 260 (2003) 249-253]. We also prove a new identity between the number of acyclic non-crossing graphs with n vertices and k edges (isolated vertices allowed and no multiple edges allowed), and the number of non-crossing connected graphs with n edges and k vertices (multiple edges allowed and no isolated vertices allowed).     

The most vital nodes with respect to independent set and vertex cover

Given an undirected graph with weights on its vertices, the k most vital nodes independent set (k most vital nodes vertex cover) problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets (minimum weight of vertex covers, respectively). We also consider the complementary problems, minimum node blocker independent set (minimum node blocker vertex cover) that consists of removing a subset of vertices of minimum size such that the maximum weight of independent sets (minimum weight of vertex covers, respectively) in the remaining graph is at most a specified value. We show that these problems are NP-hard on bipartite graphs but polynomial-time solvable on unweighted bipartite graphs. Furthermore, these problems are polynomial also on cographs and graphs of bounded treewidth. Results on the non-existence of ptas are presented, too.     

Graphs with given number of cut vertices and extremal Merrifield-Simmons index

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Denote by G(n,k) the set of connected graphs with n vertices and k cut vertices. In this paper, we characterize the graphs with the maximum and minimum Merrifield-Simmons index, respectively, among all graphs in G(n,k) for all possible k values.     

Subclasses of k-trees: Characterization and recognition

A k-tree is either a complete graph on k vertices or a graph G=(V,E) that contains a vertex whose neighbourhood in G induces a complete graph on k vertices and whose removal results in a k-tree. We present two new subclasses of k-trees and their properties. First, we present the definition and characterization of k-path graphs, based on the concept of k-paths, that generalizes the classic concept of paths. We also introduce the simple-clique k-trees, of which the maximal outerplanar graphs and the planar 3-trees are particular cases. Based on Characterization Theorems, we show recognition algorithms for both families. Finally, we establish the inclusion relations among these new classes and k-trees.