Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.     

Estimating Second-Order Characteristics of Inhomogeneous Spatio-Temporal Point Processes

Non-parametric estimates of the K-function and the pair correlation function play a fundamental role for exploratory and explanatory analysis of spatial and spatio-temporal point patterns. These estimates usually require information from outside of the study region, resulting to the so-called edge effects which have to be corrected. They also depend on first-order characteristics, which have to be estimated in practice. In this paper, we extend classical edge correction methods to the spatio-temporal setting and compare the performance of the related estimators for stationary/non-stationary and/or isotropic/anisotropic point patterns. Further, we explore the influence of the estimated intensity function on these estimators.     

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.     

Estimation of flows in flow networks

Let G be a directed graph with an unknown flow on each edge such that the following flow conservation constraint is maintained: except for sources and sinks, the sum of flows into a node equals the sum of flows going out of a node. Given a noisy measurement of the flow on each edge, the problem we address, which we call the Most Probable Flow Estimation problem (MPFE), is to estimate the most probable assignment of flow for every edge such that the flow conservation constraint is maintained. We provide an algorithm called ΔY-mpfe for solving the MPFE problem when the measurement error is Gaussian (Gaussian-MPFE). The algorithm works in O(∣E∣ + ∣V∣²) when the underlying undirected graph of G is a 2-connected planar graph, and in O(∣E∣ + ∣V∣) when it is a 2-connected serial-parallel graph or a tree. This result is applicable to any Minimum Cost Flow problem for which the cost function is τ_e(X_e − μ_e)² for edge e where μ_e and τ_e are constants, and X_e is the flow on edge e. We show that for all topologies, the Gaussian-MPFE’s precision for each edge is analogous to the equivalent resistance measured in series to this edge in an electrical network built by replacing every edge with a resistor reflecting the measurement’s precision on that edge.     

The edge L-function of a graph

We extend Watanabe and Fukumizu’s Theorem on the edge zeta function to a regular covering of a graph G. Next, we define an edge L-function of a graph G, and give a determinant expression of it. As a corollary, we present a decomposition formula for the edge zeta function of a regular covering of G by a product of edge L-functions of G.     

Total Edge Irregularity Strength of Toroidal Fullerene

A toroidal fullerene (toroidal polyhex) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. An edge irregular total k-labeling of a graph G is such a labeling of the vertices and edges with labels 1, 2, … , k that the weights of any two different edges are distinct, where the weight of an edge is the sum of the label of the edge itself and the labels of its two endvertices. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength, tes(G). In this paper we determine the exact value of the total edge irregularity strength of toroidal polyhexes.     

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.     

On 3-restricted edge connectivity of undirected binary Kautz graphs

An edge cut of a connected graph is m-restricted if its removal leaves every component having order at least m. The size of minimum m-restricted edge cuts of a graph G is called its m-restricted edge connectivity. It is known that when m≤4, networks with maximal m-restricted edge connectivity are most locally reliable. The undirected binary Kautz graph UK(2,n) is proved to be maximal 2- and 3-restricted edge connected when n≥3 in this work. Furthermore, every minimum 2-restricted edge cut disconnects this graph into two components, one of which being an isolated edge.     

Edge domatic numbers of complete n- partite graphs

An edge dominating set of a graph is a set of edgesD such that every edge not inD is adjacent to an edge inD. An edge domatic partition of a graph G =(V, E) is a collection of pairwise disjoint edge dominating sets of G whose union isE. The maximum size of an edge domatic partition of G is called the edge domatic number of G. In this paper we study the edge domatic numbers of completen-partite graphs. In particular, we give exact values for the edge domatic numbers of complete 3-partite graphs and balanced complete n-partite graphs with oddn.     

On the 3-restricted edge connectivity of permutation graphs

An edge cut W of a connected graph G is a k-restricted edge cut if G−W is disconnected, and every component of G−W has at least k vertices. The k-restricted edge connectivity is defined as the minimum cardinality over all k-restricted edge cuts. A permutation graph is obtained by taking two disjoint copies of a graph and adding a perfect matching between the two copies. The k-restricted edge connectivity of a permutation graph is upper bounded by the so-called minimum k-edge degree. In this paper some sufficient conditions guaranteeing optimal k-restricted edge connectivity and super k-restricted edge connectivity for permutation graphs are presented for k=2,3.     

Upper bounds on vertex distinguishing chromatic index of some Halin graphs

A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by ???? _s (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with ??(G) ?? 4, respectively.     

Some notes on the spectral perturbations of the signless Laplacian of a graph

Let G be a simple graph and let Q（G） be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q（G） under an edge addition or an edge contraction.     

Contractible edges in minimally k-connected graphs

An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. In this paper, we prove that a (K₁ + C₄)-free minimally k-connected graph has a k-contractible edge, if around each vertex of degree k, there is an edge which is not contained in a triangle. This implies previous two results, one due to Thomassen and the other due to Kawarabayashi.     

The Maximum Number of Edges in Geometric Graphs with Pairwise Virtually Avoiding Edges

Let G be a geometric graph on n vertices that are not necessarily in general position. Assume that no line passing through one edge of G meets the relative interior of another edge. We show that in this case the number of edges in G is at most 2n?3.     

Graphic matroids,shellability and the Poincare Conjecture

In this paper we introduce a theory of edge shelling of graphs. Whereas the standard notion of shelling a simplicial complex involves a sequential removal of maximal simplexes, edge shelling involves a sequential removal of the edges of a graph. A necessary and sufficient condition for edge shellability is given in the case of 3-colored graphs, and it is conjectured that the result holds in general. Questions about shelling, and the dual notion of closure, are motivated by topological problems. The connection between graph theory and topology is by way of a complex ΔG associated with a graph G. In particular, every closed 2- or 3-manifold can be realized in this way. If ΔG is shellable, then G is edge shellable, but not conversely. Nevertheless, the condition that G is edge shellable is strong enough to imply that a manifold ΔG must be a sphere. This leads to completely graph-theoretic generalizations of the classical Poincaré Conjecture.     

Removable edges of cycles in 5-connected graphs

Let G be a 5-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G?e; second, for each vertex x of degree 4 in G?e, delete x from G?e and then completely connect the 4 neighbors of x by K ₄. If multiple edges occur, we use single edge to replace them. The final resultant graph is denoted by G ? e. If G ? e is still 5-connected, then e is called a removable edge of G. In this paper, we investigate the distribution of removable edges in a cycle of a 5-connected graph. And we give examples to show some of our results are best possible in some sense.     

The factorial representation of balanced labelled graphs

Consider a graph G with m edges and m+1 vertices. Label the vertices of G from 0 to m. Label each edge with the positive difference of its end labels. If these edge labels take on the values from 1 to m, the graph is said to be properly labelled. If, in addition, the end labels x,y of each edge satisfy x ? r < y for some fixed integer r, the proper labelling is balanced. We introduce sequences of integers to represent properly labelled graphs. Using these sequences, we count balanced labelled graphs as well as a subclass of these which possess symmetry.     

Acyclic edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

On diameter critical graphs

A graph G is diameter k-critical if the graph has diameter k and the deletion of any edge increases its diameter. We show that every diameter 2-critical graph on v vertices has (i) at most 0.27v² edges, and (ii) average edge degree at most $\text{6}\text{5}$v. We also make a conjecture on the maximal number of edges in a diameter k-critical graph.