Edge Set Of Graph Research Articles

Distance-Edge-Monitoring Sets in Hierarchical and Corona Graphs

Let [Formula: see text] and [Formula: see text] be the vertex set and edge set of graph [Formula: see text]. Let [Formula: see text] be the distance between vertices [Formula: see text] and [Formula: see text] in the graph [Formula: see text] and [Formula: see text] be the graph obtained by deleting edge [Formula: see text] from [Formula: see text]. For a vertex set [Formula: see text] and an edge [Formula: see text], let [Formula: see text] be the set of pairs [Formula: see text] with a vertex [Formula: see text] and a vertex [Formula: see text] such that [Formula: see text]. A vertex set [Formula: see text] is distance-edge-monitoring set, introduced by Foucaud, Kao, Klasing, Miller, and Ryan, if every edge [Formula: see text] is monitored by some vertex of [Formula: see text], that is, the set [Formula: see text] is nonempty. In this paper, we determine the smallest size of distance-edge-monitoring sets of hierarchical and corona graphs.

Developing A Secure Cryptosystem with Rainbow Vertex Antimagic Coloring of Cycle Graph

An edge labeling of graph G is a function g from the edge set of graph G to the first natural numbers up to the number of the edge set. Graph G admits a rainbow vertex antimagic coloring if, for any two vertices, there is a path with different colors of all internal vertices. The vertex color of graph G is assigned by vertex weight. The vertex weight of graph G is obtained by summing all edge labels that incident with that vertex. The rainbow vertex antimagic connection number of graph G, denoted by rvac(G) is the smallest number of different colors induced by rainbow vertex antimagic coloring. In this research, we determine the upper bound of the rainbow vertex antimagic connection number (rvac) on a cycle graph (Cn) and create a secured cryptosystem using a modified Affine Cipher based on rainbow vertex antimagic coloring.

The reflexive edge strength on some almost regular graphs

A function f with domain and range are respectively the edge set of graph G and natural number up to ke, and a function f with domain and range are respectively the vertex set of graph G and the even natural number up to 2kv are called a total k-labeling where k=max{ke,2kv}. The total k-labeling of graph G by the condition that every two different edges have different weight is called an edge irregular reflexive k-labeling, where for any edge x1x2, the weight is wt(x1x2)=fv(x1)+fe(x1x2)+fv(x2). The reflexive edge strength of the graph G, denoted by res(G) is the minimum k for graph G which has an edge irregular reflexive k-labelling. In this study, we obtained the res(G) of graphs which their vertex degrees show an almost regularity properties.

Formula omitted]-circular matroids of graphs

In the 1930s Hassler Whitney considered and completely solved the problem (WP) of describing the classes of graphs G having the same cycle matroid M(G) Whitney (1932, 1933). A natural analog (WP)′ of Whitney’s problem (WP) is to describe the classes of graphs G having the same matroid M′(G), where M′(G) is a matroid (on the edge set of G) distinct from M(G). For example, the corresponding problem (WP)′=(WP)θ for the so-called bicircular matroid Mθ(G) of graph G was solved in Coullard et al. (1991) and Wagner (1985). We define the so-called k-circular matroidMk(G) on the edge set of graph G for any integer k≥0 so that M(G)=M0(G) and Mθ(G)=M1(G). We give a characterization and establish some important properties of the k-circular matroid Mk(G). The results of this paper are used in our paper De Jesús and Kelmans (2017) to study a particular problem of (WP)k on graphs uniquely defined by their k-circular matroids.

Strong Games Played on Random Graphs

In a strong game played on the edge set of a graph $G$ there are two players, Red and Blue, alternating turns in claiming previously unclaimed edges of $G$ (with Red playing first). The winner is the first one to claim all the edges of some target structure (such as a clique $K_k$, a perfect matching, a Hamilton cycle, etc.). In this paper we consider strong games played on the edge set of a random graph $G\sim G(n,p)$ on $n$ vertices. We prove that $G\sim G(n,p)$ is typically such that Red can win the perfect matching game played on $E(G)$, provided that $p\in(0,1)$ is a fixed constant.

Strong Domination Number of Some Graphs

In this paper, we consider strong domination number of P n k and C n k . Let G=(V,E) be a graph, V is a vertex set and E is an edge set of graph G and u,v ∈V . u strongly dominates v and v weakly dominates u if (i) uv ∈ E and (ii) d ( u, G ) ≥ d ( v, G ) . A set D ⊂V is a strong-dominating set ( sd -set) of G if every vertex in V-D is strongly dominated by at least one vertex in D . The strong domination number γ s of G is the minimum cardinality of an sd -set.

CHROMATICITY OF WHEELS WITH MISSING THREE SPOKES

All graph consider here are simple graphs. For a graph G, Let V(G) and E(G) be the vertex set and a edge set of graph G, respectively. The order of G is denoted by v(G), and the size of G by e(G), i.e. v(G)=|V(G)| and e(G) = |E(G)|. Let P(G,λ) (or simply P(G)) denote the chromatic polynomial of graph G. Two graphs G and H are called chromatically equivalent if P(G)=P(H), and G is called chromatically unique if P(G)=P(H) implies H isomorphic to G for any graph H [9]. A wheel W_n is a graph obtained by taking the join of K_1 and the cycle C_(n-1), edges which join K_1 to the vertices of C_(n-1) are called the spokes [2]. Let W_n be wheel of order n and let W(n,k) be the graph obtained from W_n by deleting all but k consecutive spokes, where n≥4 and 1≤k≤ n – 1. Chia [2] showed that W(n,n-2) is chromatically unique for any even integers n≥6. In [1], W(5,3) was proved to be chromatically unique. Dong and Li, [5], proved that for any odd integer n≥9,W(n,n-2) is chromatically unique, and just one graph, ( shown in Fig.1(b)) is chromatically equivalent to W(7,5), and is not isomorphic to it. It is easy to check that W(4,1) and W(5,2) are chromatically unique.

Tension continuous maps—Their structure and applications

We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly TT mappings). The existence of a TT mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see Hell and Nešetřil (2004) [10]). In this paper we study the relationship of the homomorphism and TT orders. We stress the similarities and the differences in both deterministic and random settings. Particularly, we prove that TT order is universal and investigate graphs for which homomorphisms and TT mappings coincide (so-called homotens graphs). In the course of our study, we prove a new Ramsey-type theorem, which may be of independent interest. We solve a problem asked in [Matt DeVos, Jaroslav Nešetřil, André Raspaud, On edge-maps whose inverse preserves flows and tensions, in: J.A. Bondy, J. Fonlupt, J.-L. Fouquet, J.-C. Fournier, J.L. Ramirez Alfonsin (Eds.), Graph Theory in Paris: Proceedings of a Conference in Memory of Claude Berge, in: Trends in Mathematics, Birkhäuser, 2006].

Equitable Partitions to Spanning Trees in a Graph

In this paper we first prove that if the edge set of an undirected graph is the disjoint union of two of its spanning trees, then for every subset $P$ of edges there exists a spanning tree decomposition that cuts $P$ into two (almost) equal parts. The main result of the paper is a further extension of this claim: If the edge set of a graph is the disjoint union of two of its spanning trees, then for every stable set of vertices of size 3, there exists such a spanning tree decomposition that cuts the stars of these vertices into (almost) equal parts. This result fails for 4 instead of 3. The proofs are elementary.

Binary codes from the complements of the triangular graphs

For n ≥ 4, the complement of the triangular graph denoted by T(n) is the graph of which the vertex-set is Ω{2}, the set of all 2-subsets of Ω={1,2,…,n}, and any two vertices u and v constitute an edge [u, v] if and only if u⊓v=φ. In this paper the binary codes generated by the rows of the adjacency matrix of T(n) and their corresponding duals are studied. These codes are viewed as subspaces of vector spaces defined on the edge-sets of graphs of order n. Automorphism groups are obtained for these codes, and in the non-trivial cases permutation decoding (PD)-sets of the order of n when n ≡ 1 (mod 4), and of the order of n 2 when n ≡ 3 (mod 4) are obtained for the dual code.

Cutting up graphs revisited – a short proof of Stallings' structure theorem

This is a short proof of the existence of finite sets of edges in graphs with more than one end, such that after removing them we obtain two components which are nested with all their isomorphic images. This was first done in “Cutting up graphs” [Dunwoody, Combinatorica 2: 15–23, 1982]. Together with a certain tree construction and some elementary Bass–Serre theory this yields a combinatorial proof of Stallings' theorem on the structure of finitely generated groups with more than one end.

On tension-continuous mappings

Tension-continuous (shortly TT ) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. At the same time, tension-continuous mappings are a dual notion to flow-continuous mappings, and the context of nowhere-zero flows motivates several questions considered in this paper. Extending our earlier research we define new constructions and operations for graphs (such as graphs Δ M ( G ) ) and give evidence for the complex relationship of homomorphisms and TT mappings. Particularly, solving an open problem, we display pairs of TT -comparable and homomorphism-incomparable graphs with arbitrarily high connectivity. We give a new (and more direct) proof of density of TT order and study graphs such that TT mappings and homomorphisms from them coincide; we call such graphs homotens. We show that most graphs are homotens, on the other hand every vertex of a nontrivial homotens graph is contained in a triangle. This provides a justification for our construction of homotens graphs.

On the edge set of graphs of lattice paths

This note explores a new family of graphs defined on the set of paths of the m × n lattice. We let each of the paths of the lattice be represented by a vertex, and connect two vertices by an edge if the corresponding paths share more than k steps, where k is a fixed parameter 0 = k = m + n. Each such graph is denoted by G(m, n, k). Two large complete subgraphs of G(m, n, k) are described for all values of m, n, and k. The size of the edge set is determined for n = 2, and a complicated recursive formula is given for the size of the edge set when k = 1.

Complexes of Directed Graphs

Let be a monotone property of directed graphs on n vertices, and let $\Delta_n^{P}$ denote the abstract simplicial complex whose simplices are the edge sets of graphs having property P. We prove the following: If P = acyclic,'' then $\Delta_n^{P}$ is homotopy equivalent to the (n-2)-sphere. If P = not strongly connected,'' then $\Delta_n^{P}$ has the homotopy type of a wedge of (n-1)! spheres of dimension 2n-4. The lattice of all posets on {1,2,...,n} plays an important role in the analysis. We also discuss some other properties of directed graphs from this point of view.

On matroids on edge sets of graphs with connected subgraphs as circuits II

Matroidal families are defined as families of connected graphs such that, given any graph G, the subgraphs of G isomorphic to a member of the family are the circuits of a matroid on the edge set of G. It will be proved that there are four matroidal families with all members having less than three independent cycles (Theorems 1 and 2) and that all members of any other matroidal family have at least three independent cycles (Theorem 3). The members of these four families are the complete graph on two points, the cycles, the connected graphs with two independent cycles and no pendant edges (which we call the bicircular graphs), and the members of a family formed by the even cycles and the bicircular graphs with no even cycle, respectively. Concerning any other matroidal families, we prove that no graph in such a family has a vertex of degree two (Theorem 5) and consequently that no two graphs in such a family are homeomorphic (Theorem 6).

On Matroids on Edge Sets of Graphs with Connected Subgraphs as Circuits

On Matroids on Edge Sets of Graphs with Connected Subgraphs as Circuits

On matroids on edge sets of graphs with connected subgraphs as circuits

It is proved that if F \mathcal {F} is a finite family of connected, finite graphs, then a graph G G exists such that the subgraphs of G G isomorphic to a member of the family cannot be regarded as the circuits of a matroid on the edge set of G G .