Pitchfork pendant domination in graphs

The concept of domination in graphs is an inevitable topic in the field of graph theory. The sources of domination can be seen in a wide range of real-world situations, such as radio broadcasting, computer communication networks, school bus routing, electrical power networks, influence in social networks, surveying, resource allocation, and even transporting hazardous materials. A dominating set (DS) of a finite, simple, connected, and undirected graph <img src=image/13441473_01.gif>, or simply <img src=image/13441473_02.gif> with a non-empty node set <img src=image/13441473_03.gif> and line set <img src=image/13441473_04.gif>, is a set <img src=image/13441473_05.gif> such that every node, <img src=image/13441473_06.gif> is adjacent to a node, <img src=image/13441473_07.gif>. The domination number (DN) of <img src=image/13441473_02.gif>, represented by <img src=image/13441473_08.gif>, is the cardinality of a minimum DS. Similarly, set <img src=image/13441473_09.gif> of nodes is called an independent set (IS) if no two nodes in <img src=image/13441473_10.gif> are adjacent to each other. An independent dominating set (IDS) of <img src=image/13441473_02.gif> is a set <img src=image/13441473_11.gif> that is both dominating and independent in <img src=image/13441473_02.gif>. The independent domination number (IDN) of <img src=image/13441473_02.gif>, represented by <img src=image/13441473_12.gif>, is the cardinality of a minimum IDS. Further, let <img src=image/13441473_02.gif> be with node set <img src=image/13441473_13.gif> where each <img src=image/13441473_14.gif> is a set of nodes having at least two nodes of the same degree and <img src=image/13441473_15.gif>. The degree splitting graph (DSG) of <img src=image/13441473_02.gif>, denoted by <img src=image/13441473_16.gif>, is formed from <img src=image/13441473_02.gif> by adding nodes <img src=image/13441473_17.gif> and joining to each node of <img src=image/13441473_14.gif> for <img src=image/13441473_18.gif>. In this paper, we derive the IDN of certain graphs such as lotus graph, line graph of sunlet graph, butterfly graph, shipping graph, ladder graph, fan graph, double fan graph, and their degree splitting graphs.

We use the finite dimensional monotonicity methods in order to investigate problems connected with the discrete -Laplacian on simple, connected, undirected, weighted, and finite graphs with nonlinearities given in a non-potential form. Positive solutions are also considered.

In this paper we derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the uniform multicommodity flow problem under random walk routing. We show the mean load is linearly dependent on the nodal degree with a common multiplier equal to the sum of the inverses of the non-zero eigenvalues of the graph Laplacian. Even though some aspects of the mean load value, such as linear dependence on the nodal degree, are intuitive and may be derived from the equilibrium distribution of the random walk on the undirected graph, the exact expression for the mean load in terms of the full spectrum of the graph has not been known before. Using the explicit expression for the mean load, we give asymptotic estimates for the load on a variety of graphs whose spectral density are well known. We conclude with numerical computation of the mean load for other well-known graphs without known spectral densities.

Similarity is an important way of categorizing mathematical objects. In this paper we consider how similarity between undirected graphs, or networks, can be defined. We do this by reducing any undirected graph, G = (V,E) to its irreducible spine, I. First we show that the reduction process, ω, is a unique transformation and thus is a well-defined function. Each irreducible spine, I, thus defines an equivalence class consisting of all graphs {G_i} such that G_i.ω = I. We then show that reduction by ω preserves many key properties. Specifically it preserves shortest path lengths in I, and (under rather general conditions) preserves the location of distance and betweeness centers. In essence, every member graph of the equivalence class will have much the same path structure. Next, an algorithm is presented which, given any irreducible spine I, will generate (randomly) a graph Gk in the equivalence class defined by I, that is, we must have G_k.ω = I. Since I.e.ω = I, e = ω^-1. It is of some importance in network analysis to be able to quantify the notion of network similarity; to be able to say that a network G1 is closer to the network G2 than to G3. A popular method of comparing similarity between two networks G1 and G2 is to calculate the Spearman coefficient based on corresponding node degrees. We do this for 5 random graphs in the equivalence class of our running example. Finally, we show that any irreducible spine consists of a system of cycles. We then note that every spine, I, is topologically equivalent to a cubic graph. Thus individual cubic graphs generate a second, and coarser, equivalence relation on the family of all undirected graphs, which we call its genus.

We study constraint satisfaction problems (CSPs) in the presence of counting quantifiers ∃ ≥ j , asserting the existence of j distinct witnesses for the variable in question. As a continuation of our previous (CSR 2012) paper [11], we focus on the complexity of undirected graph templates. As our main contribution, we settle the two principal open questions proposed in [11]. Firstly, we complete the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, placing each case either in P, NP-complete or Pspace-complete. This involves resolution of the cases in which we have the single quantifier ∃ ≥ j on the clique \(\mathbb{K}_{2j}\). Secondly, we confirm a conjecture from [11], which proposes a full dichotomy for ∃ and ∃ ≥ 2 on all finite undirected graphs. The main thrust of this second result is the solution of the complexity for the infinite path which we prove is a polynomial-time solvable problem. By adapting the algorithm for the infinite path we are then able to solve the problem for finite paths, and then trees and forests. Thus as a corollary to this work, combining with the other cases from [11], we obtain a full dichotomy for ∃ and ∃ ≥ 2 quantifiers on finite graphs, each such problem being either in P or NP-hard. Finally, we persevere with the work of [11] in exploring cases in which there is dichotomy between P and Pspace-complete, and contrast this with situations in which the intermediate NP-completeness may appear.

The aim of the paper is to study and to solve finite systems of equations over finite undirected graphs. Equations over graphs are atomic formulas of the language L consisting of the set of constants (graph vertices), the binary vertex adjacency predicate and the equality predicate. It is proved that the problem of checking compatibility of a system of equations S with k variables over an arbitrary simple n-vertex graph Γ is N P-complete. The computational complexity of the procedure for checking compatibility of a system of equations S over a simple graph Γ and the procedure for finding a general solution of this system is calculated. The computational complexity of the algorithm for solving a system of equations S with k variables over an arbitrary simple n-vertex graph Γ involving these procedures is O(k 2n k/2+1(k + n) 2 ) for n &gt; 3. Polynomially solvable cases are distinguished: systems of equations over trees, forests, bipartite and complete bipartite graphs. Polynomial time algorithms for solving these systems with running time O(k 2n(k + n) 2 ) are proposed.

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

A linear algorithm for the domination number of a series-parallel graph

Let G = (V, E) be a simple, finite and undirected graph and without isolated vertex. A set D ⊆ V is said to be chromatic total dominating set of G if D is a total dominating set and χ(<D>) = χ(G). The minimum cardinality of a chromatic total dominating set of G is called the chromatic total domination number of G and is denoted by γtch(G). In this paper, we discuss the chromatic total domination number for standard graphs.

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local structural properties as this finite graph, when observed by a nonbacktracking walker. Formally excluding the boundary, this infinite tree is a generalization of the Bethe lattice. We indicate an infinite, locally treelike, random network whose local structure is exactly given by this infinite tree. Message passing equations for various cooperative models on this construction are the same as for the original finite graph, but here they provide the exact solutions of the corresponding cooperative problems. These solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated. We show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix. As representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction, the nonbacktracking expansion. We analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations. Notably, in a range of important cases, simulations turn out to be more efficient computationally than the message passing.

For a positive integer $k$, a subset $D$ of vertices in a digraph $\overrightarrow{G}$ is a $k$-dominating set if every vertex not in $D$ has at least $k$ direct predecessors in $D.$ The $k$-domination number is the minimum cardinality among all $k$-dominating sets of $\overrightarrow{G}$. The game $k$-domination number of a simple and undirected graph is defined by the following game. Two players, $\mathcal{A}$ and $\mathcal{D}$, orient the edges of the graph alternately until all edges are oriented. Player $\mathcal{D}$ starts the game, and his goal is to decrease the $k$-domination number of the resulting digraph, while $\mathcal{A}$ is trying to increase it. The game $k$-domination number of the graph $G$ is the $k$-domination number of the directed graph resulting from this game. This is well defined if we suppose that both players follow their optimal strateries. We are mainly interested in the study of the game $2$-domination number, where some upper bounds will be presented. We also establish a Nordhaus-Gaddum bound for the game $2$-domination number of a graph and its complement.

In this paper we shall demonstrate that random graphs satisfy some interesting Ramsey type properties. This paper deals with finite, simple and undirected graphs only. If G and H are graphs, write G → H to mean that if the edges of G are coloured by two colours, then G contains a monochromatic copy of H.

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

LetG= (V, E)be a finite, simple and undirected graph with a vertex setVand an edge setE. An edge irregular totalk-labelling is a functionf : V ᴗE → {1,2,…,k}such that for any two different edgesxyandx’y’inE, their weights are distinct. The weight of edgexyis the sum of label of edgexy, labels of vertexxand of vertexy. The minimumkfor which the graphGadmits an edge irregular totalk-labelling is called the total edge irregularity strength ofG, denoted bytes(G). We have determined the total edge irregularity strength of book graphs, double book graphs and triple book graphs. In this paper, we show the exact value of the total edge irregularity strength of quadruplet book graphs and quintuplet book graphs.