Stabilizing the distinguishing number of a graph

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 L(2, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) - f(y)| ≥ 2 if dist(x, y) = 1 and |f(x) - f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph G. The span of an L(2, 1)-labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labelling with span at most Δ2 + 1. By settling this conjecture for Δ sufficiently large, we prove the existence of a constant C such that the span of any graph of maximum degree Δ is at most Δ 2 + C.

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake 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

Gallai-Ramsey numbers for fans

Extra connectivity of networks modeled by the strong product graphs

The vertex connectivity of a graph is the minimum number of vertices whose removal disconnects the graph, and it is a fundamental notion that measures the level of interconnectedness of vertices. Mining subgraphs with high vertex connectivity has wide applications in robust and efficient communication networks and the organization of task groups. However, existing approaches often neglect size constraints, leading to subgraphs that are either too large or too small for practical use. Therefore, in this paper, we introduce the size constraint and formulate the graph downsizing problem: Given a graph G with n vertices and a positive integer $$\tau &lt;n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> , find an induced subgraph of size $$n-\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> </mml:math> that maximizes vertex connectivity. We first analyze the computational complexity of the graph downsizing problem and show that the problem is hard in several aspects. To solve the problem in practice, we use a nesting tree structure to represent relations between maximal cohesive subgraphs in G . This structure allows us to develop an efficient heuristic, called inundation downsizing algorithm, to find highly vertex-connected subgraphs of different sizes. Experimental results reveal that our heuristic is able to produce almost optimal solutions on real-world datasets with up to 1 million vertices while incurring small time costs.

Assigning an integer to a vertices or edges is called a vertex or edge labeling respectively. Suppose G is a simple graph. Consider the function f for the edge set . For each vertex t , define f(t)= (mod2). The function f is known as an E-cordial labeling (E-CL) of G if , and where , and , are the number of edges and vertices labeled with 0 and labeled by 1 respectively. A graph that admits E-CL is said to be E-cordial graphs (E-CG). Based on the above definition we propose a novel labeling known as SP Mean E-cordial labeling (E-CL). In this paper, we study SP Mean E-CL of several families of graphs such as complete bipartite graphs, complete graphs, wheels, etc.

For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. The concept of $\lambda$-choosability puts $ k$-colourability and $k$-choosability in the same framework: If $\lambda = \{k\}$, then $\lambda$-choosability is equivalent to $k$-choosability; if $\lambda$ consists of $k $ copies of $1$, then $\lambda$-choosability is equivalent to $k $-colourability. %We say $G$ is non-chromatic-$\lambda$-choosable if $\chi(G)=k_{\lambda}$ and $G$ is not $\lambda$-choosable. If $G$ is $\lambda$-choosable, then $G$ is $k_{\lambda}$-colourable. On the other hand, there are $k_{\lambda}$-colourable graphs that are not $\lambda$-choosable, provided that $\lambda$ contains an integer larger than $1$. Let $\phi(\lambda)$ be the minimum number of vertices in a $k_{\lambda}$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\phi(\lambda)$ for all $\lambda$.

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$. We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.

The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of the join of two graphs G and H, i.e., G+H. We prove that 0≤ D(G+H)-max{D(G),D(H)}≤ z, where z depends on the number of some induced subgraphs generated by some suitable partitions of V(G) and V(H). Let Gk be the k-th power of G with respect to the join product. We prove that if G is a connected graph of order n ≥ 2, then Gk has the distinguishing index 2, except D'(K2+K2)=3.

The distinguishing number (index) of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling (edge labeling) with $d$ labels that is preserved only by the trivial automorphism. A set $S$ of vertices in $G$ is a dominating set of $G$ if every vertex of $V(G)setminus S$ is adjacent to some vertex in $S$. The minimum cardinality of a dominating set of $G$ is the domination number of $G$. In this paper, we obtain some upper bounds for the distinguishing number and the distinguishing index of a graph based on its domination number.