Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

Lower bounds on the signed domination numbers of directed graphs

A numerical invariant of directed graphs concerning domination which is named signed domination number γ_S is studied in this paper. We present some sharp lower bounds for γ_S in terms of the order, the maximum degree and the chromatic number of a directed graph.     

Minimum sum edge colorings of multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.     

Minus domination in small-degree graphs

Minus domination in graphs is a variant of domination where the vertices must be labeled −1,0,+1 such that the sum of labels in each N[v] is positive. (As usual, N[v] means the set containing v together with its neighbors.) The minus domination number γ⁻ is the minimum total sum of labels that can be achieved. In this paper we prove linear lower bounds for γ⁻ in graphs either with Δ⩽3, or with Δ⩽4 but without vertices of degree 2. The central section is concerned with complexity results for Δ⩽4: We show that computing γ⁻ is NP-hard and MAX SNP-hard there, but that γ⁻ can be approximated in linear time within some constant factor. Finally, our approach also applies to signed domination (where the labels are −1,+1 only) in small-degree graphs.     

Upper minus total domination in small-degree regular graphs

A function f:V(G)→{-1,0,1} defined on the vertices of a graph G is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. An MTDF f is minimal if there does not exist an MTDF g:V(G)→{-1,0,1}, f≠g, for which g(v)?f(v) for every v∈V(G). The weight of an MTDF is the sum of its function values over all vertices. The minus total domination number of G is the minimum weight of an MTDF on G, while the upper minus domination number of G is the maximum weight of a minimal MTDF on G. In this paper we present upper bounds on the upper minus total domination number of a cubic graph and a 4-regular graph and characterize the regular graphs attaining these upper bounds.     

Total Colorings of Graphs with Minimum Sum of Colors

The total chromatic sum of a graph is the minimum sum of colors (natural numbers) taken over all proper colorings of vertices and edges of a graph. We construct infinite families of graphs for which the minimum number of colors to achieve the total chromatic sum is larger than the total chromatic number.     

A decomposition approach for solving a broadcast domination network design problem

We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers f _i assigned to each node i, such that for any node j in the graph, there exists some node k having a positive f _k -value whose shortest distance to node j is no more than f _k . The cost of a broadcast domination solution is the sum of all node power values. The network design problem constructs edges that decrease the minimum broadcast domination cost on the graph. The overall problem we consider minimizes the sum of edge construction costs and broadcast domination costs. We show that this problem is NP-hard in the strong sense, even on unweighted graphs. We then propose a decomposition strategy, which iteratively adds valid inequalities based on optimal broadcast domination solutions corresponding to the first-stage network design solutions. We demonstrate that our decomposition approach is computationally far superior to the solution of a single large-scale mixed-integer programming formulation.     

Total domination stable graphs upon edge addition

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. A graph is total domination edge addition stable if the addition of an arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge addition stable graphs. We determine a sharp upper bound on the total domination number of total domination edge addition stable graphs, and we determine which combinations of order and total domination number are attainable. We finish this work with an investigation of claw-free total domination edge addition stable graphs.     

Total domination critical and stable graphs upon edge removal

A set S of vertices in a graph G is a total dominating set of G 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. A graph is total domination edge critical if the removal of any arbitrary edge increases the total domination number. On the other hand, a graph is total domination edge stable if the removal of any arbitrary edge has no effect on the total domination number. In this paper, we characterize total domination edge critical graphs. We also investigate various properties of total domination edge stable graphs.     

Total domination changing and stable graphs upon vertex removal

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. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.     

Convex domination in the composition and cartesian product of graphs

In this paper we characterize the convex dominating sets in the composition and Cartesian product of two connected graphs. The concepts of clique dominating set and clique domination number of a graph are defined. It is shown that the convex domination number of a composition G[H] of two non-complete connected graphs G and H is equal to the clique domination number of G. The convex domination number of the Cartesian product of two connected graphs is related to the convex domination numbers of the graphs involved.     

On the sum of k largest singular values of graphs and matrices

In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan k-norms, given by the sum of the k largest singular values, which are studied more generally in the present paper. Several relations to chromatic number, spectral radius, spread, and to other fundamental parameters are outlined. Some results are extended to more general matrices.     

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 eccentric distance sum of graphs

The eccentric distance sum is a novel topological index that offers a vast potential for structure activity/property relationships. For a graph G, it is defined as ξd(G)=_∑v∈Vε(v)D(v), where ε(v) is the eccentricity of the vertex v and D(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. Motivated by [G. Yu, L. Feng, A. Ili?, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 934-944], in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum. Various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the Wiener index, the degree distance, eccentric connectivity index, independence number, connectivity, matching number, chromatic number and clique number are established. In addition, we present explicit formulae for the values of eccentric distance sum for the Cartesian product, applied to some graphs of chemical interest (like nanotubes and nanotori).     

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.     

On the diameter of a domination dot-critical graph

A graph G is domination dot-critical, or just dot-critical, if contracting any edge decreases the domination number. It is totally dot-critical if identifying any two vertices decreases the domination number. In this paper, we study an open question concerning of the diameter of a domination dot-critical graph G.     

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number.     

Efficient algorithms for Roman domination on some classes of graphs

A Roman dominating function of a graph G=(V,E) is a function f:V→{0,1,2} such that every vertex x with f(x)=0 is adjacent to at least one vertex y with f(y)=2. The weight of a Roman dominating function is defined to be f(V)=∑_x∈Vf(x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer an open question mentioned in [E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11-22] by showing that the Roman domination number of an interval graph can be computed in linear time. We then show that the Roman domination number of a cograph (and a graph with bounded cliquewidth) can be computed in linear time. As a by-product, we give a characterization of Roman cographs. It leads to a linear-time algorithm for recognizing Roman cographs. Finally, we show that there are polynomial-time algorithms for computing the Roman domination numbers of -free graphs and graphs with a d-octopus.     

Upper bounds for the domination subdivision and bondage numbers of graphs on topological surfaces

For a graph property P and a graph G, we define the domination subdivision number with respect to the property P to be the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to change the domination number with respect to the property P. In this paper we obtain upper bounds in terms of maximum degree and orientable/non-orientable genus for the domination subdivision number with respect to an induced-hereditary property, total domination subdivision number, bondage number with respect to an induced-hereditary property, and Roman bondage number of a graph on topological surfaces.     

Upper bounds for independent domination in regular graphs

Let G be a regular graph of order n and degree δ. The independent domination numberi(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds, as functions of n and δ, for the sum and product of the independent domination numbers of a regular graph and its complement.