On Oriented Arc-Coloring of Subcubic Graphs

There are many practical applications that require simplification of polylines. Some of the goals are to reduce the amount of information necessary to store, improve processing time, or simplify editing. The simplification is usually done by removing some of the vertices, making the resultant polyline go through a subset of the source polyline vertices. However, such approaches do not necessarily produce a new polyline with the minimum number of vertices. The approximate solution to find a polyline, within a specified tolerance, with the minimum number of vertices is described in this paper.

Constructions of bi-regular cages

On the order of [formula omitted]-cages of even girth

An outer-connected dominating set in a graph G = (V, E) is a set of vertices D ⊆ V satisfying the condition that, for each vertex v ? D, vertex v is adjacent to some vertex in D and the subgraph induced by V?D is connected. The outer-connected dominating set problem is to find an outer-connected dominating set with the minimum number of vertices which is denoted by ?~c(G)$\tilde {\gamma }_{c}(G)$. In this paper, we determine ?~c(S(n,k))$\tilde {\gamma }_{c}(S(n,k))$, ?~c(S+(n,k))$\tilde {\gamma }_{c}(S^{+}(n,k))$, ?~c(S++(n,k))$\tilde {\gamma }_{c}(S^{++}(n,k))$, and ?~c(Sn)$\tilde {\gamma }_{c}(S_{n})$, where S(n, k), S+(n, k), S++(n, k), and Sn are Sierpin?$\acute {\mathrm {n}}$ski-like graphs.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc (G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Finding minimal convex nested polygons

Odd graph and its applications to the strong edge coloring

Star Chromatic Index of Subcubic Graphs

Acyclic edge coloring of subcubic graphs

We give a complete description of the set of triplesof real numbers with the following property. There exists a constantKsuch thatis a lower bound for the matching numberof every connected subcubic graphG, wheredenotes the number of vertices of degreeifor eachi.

LetG=(V,E)be a connected simple graph and letMbe a nonempty subset ofV. TheM-distance pattern of a vertexuinGis the set of all distances fromuto the vertices inM. If the distance patterns of all vertices inVare distinct, then the setMis a distance pattern distinguishing set ofG. A graphGwith a distance pattern distinguishing set is called a distance pattern distinguishing graph. Minimum number of vertices in a distance pattern distinguishing set is called distance pattern distinguishing number of a graph. This paper initiates a study on the problem of finding distance pattern distinguishing number of a graph and gives bounds for distance pattern distinguishing number. Further, this paper provides an algorithm to determine whether a graph is a distance pattern distinguishing graph or not and hence to determine the distance pattern distinguishing number of that graph.

The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by a trivial automorphism. The distinguishing stability of a graph G is denoted by and is the minimum number of vertices whose removal changes the distinguishing number. We show that for every natural number k, there exists a graph G such that . Then we give some upper bounds for . We also study the edge distinguishing stability number (distinguishing bondage number) of G.

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem and has recently been introduced by Park and Ihm.15In this paper, we examine properties of strong matching preclusion for alternating group graphs, by finding their strong matching preclusion numbers and categorizing all optimal solutions. More importantly, we prove a general result on taking a Cartesian product of a graph with K2(an edge) to obtain the corresponding results for split-stars.

Let G=(V, E) be a graph. Let D be a minimum total dominating set of G. If V–D contains a total dominating set D’ of G, then D’ is called an inverse total dominating set with respect to D. The inverse total domination number of G is the minimum number of vertices in an inverse total dominating set of G. We initiate the study of inverse total domination in graphs and present some bounds and some exact values for . Also, some relationships between and other domination parameters are established.

Super connectivity of folded twisted crossed cubes