A Reduction of the Anticoloring Problem to Connected 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, for such colorings with many vertices colored in each color. We show that, to solve the anticoloring problem with two colors for general graphs, it suffices to solve it for connected graphs.     

Anticoloring of a family of grid 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. We deal with the anticoloring problem with two colors, in the particular cases of graphs which are strong products of two paths or two cycles, and provide an explicit optimal solution.     

Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 2-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 3-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 2-edge-colored graph H such that each graph of the considered class admits a homomorphism to H.     

Complexity of the packing coloring problem for trees

Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i-th class have pairwise distance greater than i. The main result of this paper is a solution of an open problem of Goddard et al. showing that the decision whether a tree allows a packing coloring with at most k classes is NP-complete.We further discuss specific cases when this problem allows an efficient algorithm. Namely, we show that it is decideable in polynomial time for graphs of bounded treewidth and diameter, and fixed parameter tractable for chordal graphs.We accompany these results by several observations on a closely related variant of the packing coloring problem, where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers.     

Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 2-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 3-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 2-edge-colored graph H such that each graph of the considered class admits a homomorphism to H.     

The backup 2-median problem on block graphs

The backup 2-median problem is a location problem to locate two facilities at vertices with the minimum expected cost where each facility may fail with a given probability. Once a facility fails, the other one takes full responsibility for the services. Here we assume that the facilities do not fail simultaneously. In this paper, we consider the backup 2-median problem on block graphs where any two edges in one block have the same length and the lengths of edges on different blocks may be different. By constructing a tree-shaped skeleton of a block graph, we devise an O（n log n q- m）-time algorithm to solve this problem where n and m are the number of vertices and edges, respectively, in the given block graph.     

Chromatic number of sparse colored mixed planar graphs

A colored mixed graph has vertices linked by both colored arcs and colored edges. The chromatic number of such a graph G is defined as the smallest order of a colored mixed graph H such that there exists a (arc-color preserving) homomorphism from G to H. We study in this paper the colored mixed chromatic number of planar graphs, partial 2-trees and outerplanar graphs with given girth.     

Embeddings of chemical graphs in hypercubes

We study planar graphs embedded in the plane that have chemical applications: the degrees of all vertices are 3 or 2, all internal faces but one or two arer-gons, and each internal face is a simply connected domain. For wide classes of such graphs, we solve the existence problem for embeddings of the graph metric on the vertices in multidimensional cubes or cubical lattices preserving or doubling all the distances. Incidentally we present a complete classification of some interesting families of such graphs. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 339–352, September, 2000.     

Hereditary Efficiently Dominatable Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C₄)‐free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw‐free graphs.     

R-domination on block graphs

The k-domination problem is to select a minimum cardinality vertex set D of a graph G such that every vertex of G is within distance k from some vertex of D. We consider a generalization of the k-domination problem, called the R-domination problem. A linear algorithm is presented that solves this problem for block graphs. Our algorithm is a generalization of Slater's algorithm [12], which is applicable for forest graphs.     

The capture time of a graph

We consider the game of Cops and Robbers played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to a new parameter, the capture time of a graph. While the capture time of a cop-win graph on n vertices is bounded above by n−3, half the number of vertices is sufficient for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have capture time within a small additive constant of the number of vertices. We consider the ratio of the capture time to the number of vertices, and extend this notion of capture time density to infinite graphs. For the infinite random graph, the capture time density can be any real number in [0,1]. We also consider the capture time when more than one cop is required to win. While the capture time can be calculated by a polynomial algorithm if the number k of cops is fixed, it is NP-complete to decide whether k cops can capture the robber in no more than t moves for every fixed t.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Perfect matchings after vertex deletions

This paper considers some classes of graphs which are easily seen to have many perfect matchings. Such graphs can be considered robust with respect to the property of having a perfect matching if under vertex deletions (with some mild restrictions), the resulting subgraph continues to have a perfect matching. It is clear that you can destroy the property of having a perfect matching by deleting an odd number of vertices, by upsetting a bipartition or by deleting enough vertices to create an odd component. One class of graphs we consider is the m×m lattice graph (or grid graph) for m even. Matchings in such grid graphs correspond to coverings of an m×m checkerboard by dominoes. If in addition to the easy conditions above, we require that the deleted vertices be apart, the resulting graph has a perfect matching. The second class of graphs we consider is a k-fold product graph consisting of k copies of a given graph G with the ith copy joined to the i+1st copy by a perfect matching joining copies of the same vertex. We show that, apart from some easy restrictions, we can delete any vertices from the kth copy of G and find a perfect matching in the product graph with k suitably large.     

Maximal independent sets in caterpillar graphs

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (intersection graph of mis) and the clique graph (intersection graph of cliques) of complete caterpillar graphs.     

An implicit representation of chordal comparability graphs in linear time

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O(n²).     

Heavy fans, cycles and paths in weighted graphs of large connectivity

A set of paths joining a vertex y and a vertex set L is called (y,L)-fan if any two of the paths have only y in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices.In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 3-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 2-connected weighted graphs to 3-connected weighted graphs.     

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.     

On the Minimum Sum Coloring of P 4-Sparse Graphs

In this paper, we study the minimum sum coloring (MSC) problem on P ₄-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. Based in the concept of maximal sequence associated with an optimal solution of the MSC problem of any graph, we show that there is a large sub-family of P ₄-sparse graphs for which the MSC problem can be solved in polynomial time. Moreover, we give a parameterized algorithm and a 2-approximation algorithm for the MSC problem on general P ₄-sparse graphs.     

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.