Minimum Sum Coloring of P4-sparse graphs

In this paper, we study the Minimum Sum Coloring (MSC) problem on P₄-sparse graphs. In the MSC problem, we aim to assign natural numbers to vertices of a graph such that adjacent vertices get different numbers, and the sum of the numbers assigned to the vertices is minimum. First, we introduce the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph. Next, based in such maximal sequences, we show that there is a large sub-family of P₄-sparse graphs for which the MSC problem can be solved in polynomial-time.     

Recognizing Well Covered Graphs of Families with Special P 4-Components

A graph G is called well covered if every two maximal independent sets of G have the same number of vertices. In this paper we shall use the modular and primeval decomposition techniques to decide well coveredness of graphs such that, either all their P ₄-connected components (in short, P ₄-components) are separable or they belong to well known classes of graphs that, in some local sense, contain few P ₄’s. In particular, we shall consider the class of cographs, P ₄-reducible, P ₄-sparse, extended P ₄-reducible, extended P ₄-sparse graphs, P ₄-extendible graphs, P ₄-lite graphs, and P ₄-tidy graphs.     

Minimum sum set coloring of trees and line graphs of trees

In this paper, we study the minimum sum set coloring (MSSC) problem which consists in assigning a set of x(v) positive integers to each vertex v of a graph so that the intersection of sets assigned to adjacent vertices is empty and the sum of the assigned set of numbers to each vertex of the graph is minimum. The MSSC problem occurs in two versions: non-preemptive and preemptive. We show that the MSSC problem is strongly NP-hard both in the preemptive case on trees and in the non-preemptive case in line graphs of trees. Finally, we give exact parameterized algorithms for these two versions on trees and line graphs of trees.     

Some Defective Parameters in Graphs

Given a graph G and an integer k, a set S of vertices in G is k-sparse if S induces a graph with a maximum degree of at most k. Many parameters in graph theory are defined in terms of independent sets. Accordingly, their definitions can be expanded taking into account the notion of k-sparse sets. In this discussion, we examine several of those extensions. Similarly, S is k-dense if S induces a k-sparse graph in the complement of G. A partition of V(G) is a k-defective cocoloring if each part is k-sparse or k-dense. The minimum order of all k-defective cocolorings is the k-defective cochromatic number of G and denoted z _k (G). Analogous notions are defined similarly for k-defective coloring where V(G) is partitioned into k-sparse parts. We show the NP-hardness of computing maximum k-defective sets in planar graphs with maximum degree at most k + 1 and arbitrarily large girth. We explore the extension of Ramsey numbers to k-sparse and k-dense sets of vertices. Lastly, we discuss some bounds related to k-defective colorings and k-defective cocolorings.     

Minimum Color Sum of Bipartite Graphs

The problem ofminimum color sumof a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated withinn^{1 − ε}, for any ε > 0, unlessNP = ZPP(Bar-Noyet al., Information and Computation140(1998), 183–202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unlessP = NP. The proof is byL-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.     

On vertex symmetric digraphs

The problem of how “near” we can come to a n-coloring of a given graph is investigated. I.e., what is the minimum possible number of edges joining equicolored vertices if we color the vertices of a given graph with n colors. In its generality the problem of finding such an optimal color assignment to the vertices (given the graph and the number of colors) is NP-complete. For each graph G, however, colors can be assigned to the vertices in such a way that the number of offending edges is less than the total number of edges divided by the number of colors. Furthermore, an Ω(epn) deterministic algorithm for finding such an n-color assignment is exhibited where e is the number of edges and p is the number of vertices of the graph (e?p?n). A priori solutions for the minimal number of offending edges are given for complete graphs; similarly for equicolored K_m in K_p and equicolored graphs in K_p.     

A short proof of the NP-completeness of minimum sum interval coloring

In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive different numbers and the sum of the numbers is minimized. Szkalicki has shown that minimum sum coloring is NP-hard for interval graphs. Here we present a simpler proof of this result.     

Restricted coloring problems on Graphs with few P 4’s

In this paper, we obtain linear time algorithms to determine the acyclic chromatic number, the star chromatic number, the non repetitive chromatic number and the clique chromatic number of P ₄-tidy graphs and (q, q ? 4)-graphs, for every fixed q, which are the graphs such that every set with at most q vertices induces at most q ? 4 distinct P ₄’s. These classes include cographs and P ₄-sparse graphs. We also obtain a linear time algorithm to compute the harmonious chromatic number of connected P ₄-tidy graphs and connected (q, q ? 4)-graphs. All these coloring problems are known to be NP-hard for general graphs. These algorithms are fixed parameter tractable on the parameter q(G), which is the minimum q such that G is a (q, q ? 4)-graph. We also prove that every connected (q, q ? 4)-graph with at least q vertices is 2-clique-colorable and that every acyclic coloring of a cograph is also nonrepetitive, generalizing the main result of Lyons (2011).     

Hamilton cycles in claw-heavy graphs

A graph G on n≥3 vertices is called claw-heavy if every induced claw (K_1,3) of G has a pair of nonadjacent vertices such that their degree sum is at least n. In this paper we show that a claw-heavy graph G has a Hamilton cycle if we impose certain additional conditions on G involving numbers of common neighbors of some specific pair of nonadjacent vertices, or forbidden induced subgraphs. Our results extend two previous theorems of Broersma, Ryjá?ek and Schiermeyer [H.J. Broersma, Z. Ryjá?ek, I. Schiermeyer, Dirac’s minimum degree condition restricted to claws, Discrete Math. 167-168 (1997) 155-166], on the existence of Hamilton cycles in 2-heavy graphs.     

Power domination in graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number γ_P(G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.     

The crossing numbers of join of the special graph on six vertices with path and cycle

There are only few results concerning crossing numbers of join of some graphs. In the paper, for the special graph H on six vertices we give the crossing numbers of its join with n isolated vertices as well as with the path P_n on n vertices and with the cycle C_n.     

Optimal broadcast domination in polynomial time

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that different vertices can be assigned different domination powers. Broadcast domination assigns an integer power f(v)?0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v)?1. The optimal broadcast domination problem seeks to minimize the sum of the powers assigned to the vertices of the graph. Since the presentation of this problem its computational complexity has been open, and the general belief has been that it might be NP-hard. In this paper, we show that optimal broadcast domination is actually in P, and we give a polynomial time algorithm for solving the problem on arbitrary graphs, using a non-standard approach.     

Connected graphs with a minimal number of spanning trees

The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number t_μ(G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges qis ≥(₂^P)?p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case whereq<(₂^P)?p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t₀(G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.     

The spum and sum-diameter of graphs: Labelings of sum graphs

A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph G is the minimum difference between the largest and smallest labels in a sum graph consisting of G and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the requirement that the number of additional isolated vertices in the sum graph is minimal, which we believe is a more natural quantity to study. We then provide asymptotically tight general bounds on both sides for the sum-diameter, and study its behavior under numerous binary graph operations as well as vertex and edge operations. Finally, we generalize the sum-diameter to hypergraphs.     

Path factors and parallel knock-out schemes of almost claw-free graphs

An H₁,{H₂}-factor of a graph G is a spanning subgraph of G with exactly one component isomorphic to the graph H₁ and all other components (if there are any) isomorphic to the graph H₂. We completely characterise the class of connected almost claw-free graphs that have a P₇,{P₂}-factor, where P₇ and P₂ denote the paths on seven and two vertices, respectively. We apply this result to parallel knock-out schemes for almost claw-free graphs. These schemes proceed in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is reducible if such a scheme eliminates every vertex in the graph. Using our characterisation, we are able to classify all reducible almost claw-free graphs, and we can show that every reducible almost claw-free graph is reducible in at most two rounds. This leads to a quadratic time algorithm for determining if an almost claw-free graph is reducible (which is a generalisation and improvement upon the previous strongest result that showed that there was a O(n^5.376) time algorithm for claw-free graphs on n vertices).     

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.     

2K2 vertex-set partition into nonempty parts

A graph is 2K₂-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C, D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. Determining whether an arbitrary graph is 2K₂-partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational complexity is open. We show that for C₄-free graphs, circular-arc graphs, spiders, P₄-sparse graphs, and bipartite graphs the 2K₂-partition problem can be solved in polynomial time.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.     

A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ^*/logΔ^*) of the optimum in O(mnlogn)-time, where Δ^* is the minimum k such that there is a spanning tree of the graph with maximum degree k.     

The cover pebbling number of graphs

A pebbling move on a graph consists of taking two pebbles off of one vertex and placing one pebble on an adjacent vertex. In the traditional pebbling problem we try to reach a specified vertex of the graph by a sequence of pebbling moves. In this paper we investigate the case when every vertex of the graph must end up with at least one pebble after a series of pebbling moves. The cover pebbling number of a graph is the minimum number of pebbles such that however the pebbles are initially placed on the vertices of the graph we can eventually put a pebble on every vertex simultaneously. We find the cover pebbling numbers of trees and some other graphs. We also consider the more general problem where (possibly different) given numbers of pebbles are required for the vertices.