Vertex Switching Research Articles

2-Vertex Self Switching of Trees

For a finite undirected graph \(G(V,E)\) and a non empty subset \(\sigma \subseteq V\), the switching of \(G\) by \(\sigma\) is defined as the graph \(G^\sigma (V, E')\) which is obtained from \(G\) by removing all edges between \(\sigma\) and its complement \(V\)-\(\sigma\) and adding as edges all non-edges between \(\sigma\) and \(V\)-\(\sigma\). For \(\sigma=\{v\}\), we write \(G^v\) instead of \(G^{\{v\}}\) and the corresponding switching is called as vertex switching. We also call it as \(|\sigma|\)-vertex switching. When \(|\sigma|=2\), it is termed as 2-vertex switching. If \(G \cong G^\sigma\), then it is called self vertex switching. A subgraph \(B\) of \(G\) which contains \(G[\sigma]\) is called a joint at \(\sigma\) in \(G\) if \(B-\sigma\) is connected and maximal. If \(B\) is connected, then we call \(B\) as a \(c\)-joint and otherwise a \(d\)-joint. A graph with no cycles is called an acyclic graph. A connected acyclic graph is called a tree. In this paper, we give necessary and sufficient conditions for a graph \(G\), for which \(G^{\sigma}\) at \(\sigma =\{u, v\}\) to be connected and acyclic when \(uv\in E(G)\) and \(uv \notin E(G)\). Using this, we characterize trees with a 2-vertex self switching.

Some Results on 2-Vertex Switching in Joints

For a finite undirected graph \(G(V,E)\) and a non empty subset \(\sigma\subseteq V\), the switching of \(G\) by \(\sigma\) is defined as the graph \(G^{\sigma}(V,E')\) which is obtained from \(G\) by removing all edges between \(\sigma\) and its complement \(V\)-\(\sigma\) and adding as edges all non-edges between \(\sigma\) and \(V\)-\(\sigma\). For \(\sigma = \{v\}\), we write \(G^{v}\) instead of \(G^{\{v\}}\) and the corresponding switching is called as vertex switching . We also call it as \(|\sigma |\)-vertex When \(|\sigma | = 2\), we call it as 2-vertex switching. A subgraph \(B\) of \(G\) which contains \(G[\sigma ]\) is called a joint at \(\sigma\) in \(G\) if \(B\)-\(\sigma\) is connected and maximal. If \(B\) is connected, then we call \(B\) as \(c\) -joint otherwise \(d\) -joint . In this paper, we give a necessary and sufficient condition for a \(c\)-joint \(B\) at \(\sigma = \{u,v\}\) in \(G\) to be a \(c\)-joint and a \(d\)-joint at \(\sigma\) in \(G^{\sigma}\) and also a necessary and sufficient condition for a \(d\)-joint \(B\) at \(\sigma = \{u,v\}\) in \(G\) to be a \(c\)-joint and a \(d\)-joint at \(\sigma\) in \(G^{\sigma}\) when \(uv\in E(G)\) and when \(uv\notin E(G)\).

Anti-duplication self vertex switching in some graphs

Anti-duplication self vertex switching in some graphs

Tutte’s dichromate for signed graphs

We introduce the trivariate Tutte polynomial of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the “dichromate” for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank–size generating function).The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set — for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.

2-Vertex self switching of umbrella graph

By a graph \(G=(V, E)\) we mean a finite undirected graph without loops or multiple edges. Let \(G\) be a graph and \(\sigma \subseteq V\) be a non-empty subset of \(V\). Then \(\sigma\) is said to be a self switching of \(G\) if and only if \(G \cong G^\sigma\). It can also be referred to as \(|\sigma|\)-vertex self-switching. The set of all self switching of the graph \(G\) with cardinality \(k\) is represented by \(S_k(G)\) and its cardinality by \(s s_k(G)\). A vertex \(v\) of a graph \(G\) is said to be self vertex switching if \(G \cong G^v\). The set of all self vertex switchings of \(G\) is denoted by \(\operatorname{SS}_1(G)\) and its cardinality is given by \(s s_1(G)\). If \(|\sigma|=2\), we call it as a 2-vertex self switching. The set of all 2-vertex switchings of \(G\) is denoted by \(\operatorname{SS}_2(G)\) and its cardinality is given by \(s s_2(G)\). In this paper we find the number of 2-vertex self switching vertices for the umbrella graph \(U_{m, n}\).

Inverse isolate domination number on a vertex switching of cycle related graphs

Let \(G\) be non-trivial graph. A subset \(S \subset V(G)\) is called a isolate dominating set of \(G\) if is a dominating set and \(\delta(<S>)=0\). The set \(S^{\prime} \subset V(G)-S\) such that \(S^{\prime}\) is a dominating set of \(G\) and \(\delta\left(<S^{\prime}>\right)=0\), then \(S^{\prime}\) is called an inverse isolate dominating set with respect to \(S\). The minimum cardinality of an inverse isolate dominating set is called an inverse isolate dominating number and is denoted by \(\gamma_0^{-1}(G)\). In this paper we find inverse isolate dominating number on vertex switching of some cycle related graphs.

Some results on strong 2 - vertex duplication self switching of some connected graphs

A vertex \(v \in V(G)\) is said to be a self vertex switching of \(G\) if \(G\) is isomorphic to \(G^v\), where \(G^v\) is the graph obtained from \(G\) by deleting all edges of \(G\) incident to \(v\) in \(G\) and adding all edges incident to \(v\) which are not in \(G\). A vertex \(v^{\prime}\) is the duplication of \(v\) if all the vertices which are adjacent to \(v\) in \(G\) are also adjacent to \(v^{\prime}\) in \(D(v G)\), which is the duplication graph of \(G\). Duplication self vertex switching of various graphs are given in the literature. In this paper we discuss about the 2-vertex duplication self switching graphs.

Face Sum Divisor Cordial Graphs

In this paper, we investigate the face sum divisor cordial labeling of switching of any vertex in cycle Cn, switching of a pendent vertex in path Pn and S′(K1,n).

3-Total Sum Cordial Labeling on Some New Graphs

Let \(G=(V,E)\) be a graph with vertex set \(V\) and edge set \(E\). Consider a vertex labeling \(f:V(G)\to \{0,1,2\}\) such that each edge \(uv\) assign the label \((f(u)+f(v)) (\{{\rm mod}\ 3)\). The map \(f\) is called a 3-total sum cordial labeling if \(|f(i)-f(j)|\le 1\), for \(i,j \in \{0,1,2\}\) where \(f(x)\) denotes the total number of vertices and edges labeled with \(x=\{0,1,2\}\). Any graph which satisfied 3-total sum cordial labeling is called a 3-total sum cordial graph. Here we prove some graphs like wheel, globe and a graph obtained by switching and duplication of arbitrary vertex of a cycle are 3-total sum cordial graphs.

Duplication and Switching of Divisor Cordial Graphs

Duplication and Switching of Divisor Cordial Graphs P. Maya, T. Nicholas Abstract A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, 3, . . .,|V|} such that if an edge uv is assigned the label 1 if f(u) divides f(v) or f(v) divides f(u) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. If a graph has a divisor cordial labeling, then it is called divisor cordial graph. In this paper, we prove that fan graph, switching of a pendant vertex of a helm graph, switching of a vertex of flower graph, switching of closed helm graph, and also duplication of an arbitrary vertex by an edge of a fan graph are divisor cordial. Full Text: PDF DOI: 10.15640/arms.v4n2a3

Product Cordial Graph in the Context of Some Graph Operations on Gear Graph

A graph is said to be a product cordial graph if there exists a function with each edge assign the label , such that the number of vertices with label 0 and the number of vertices with label 1 differ atmost by 1, and the number of edges with label 0 and the number of edges with label 1 differ by atmost 1. We discuss the product cordial labeling of the graphs obtained by duplication of some graph elements of gear graph. Also, we derive some product cordial graphs obtained by vertex switching operation on gear graph.

Self Vertex Switching of Connected Two-Cyclic Graphs

A vertex v ∊ V(G) is said to be a self vertex switching of G if G is isomorphic to Gv, where Gv is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. A graph G is called a two-cyclic graph if it has exactly two cycles. Trees, forests and both connected and disconnected unicyclic graphs are characterized, each with a self vertex switching. In this paper, we characterize connected two-cyclic graphs with a self vertex switching.

Self vertex switchings of connected unicyclic graphs

A vertex v ∈ V(G) is said to be a self vertex switching of G if G is isomorphic to Gv , where Gv is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. In [6], the author characterized trees each with a self vertex switching. In this paper we characterize connected unicyclic graphs each with a self vertex switching.

Some Switching Invariant Prime Graphs

We investigate prime labeling for some graphs resulted from switching of a vertex. We discuss switching invariance of some prime graphs and prove that the graphs obtained by switching of a vertex in Pn and K1,n admit prime labeling. Moreover we discuss prime labeling for the graph obtained by switching of vertex in wheel Wn.

Prime Labeling for Some Cycle Related Graphs

Here we investigate prime labeling for some cycle related graphs. We also discuss prime labeling in the context of graph operations namely fusion, duplication and vertex switching in cycle Cn.

Interchange similar self vertex switchings in graphs

A vertex v ∈ V(G) is said to be a self vertex switching of G if G is isomorphic to Gv , where Gv is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. Two vertices u and v in G are said to be interchange similar if there exists an automorphism α of G such that α(u) = v and α(v) = u. In this paper, we give a sufficient condition for a self vertex switching to be interchange similar with another vertex.

Graph Reconstruction - Some New Developments*

The reconstruction conjecture claims that every graph on at least three vertices is uniquely determined by its collection of vertex deleted subgraphs. Even though it is one of the foremost unsolved problems in Graph Theory, work on it has slowed down, may be due to the general feeling that existing techniques are not likely to lead to a complete solution. Here we give the results that have appeared recently in some selected variations of the reconstruction problem like edge reconstruction, degree associated reconstruction, vertex switching reconstruction and reconstruction numbers.

Acyclicity of Switching Classes

For a finite undirected graphG= (V,E) and a subsetA⊆V, the vertex switching ofGbyAis defined as the graphGA= (V,E'), which is obtained fromGby removing all edges betweenAand its complementAand adding as edges all nonedges betweenAandA. The switching class [G] determined byGconsists of all vertex switchingsGAfor subsetsA⊆V. We prove that the trees of a switching class [G] are isomorphic to each other. We also determine the types of treesTthat have isomorphic copies in [G]. Finally we show that apart from one exceptional type of forest, the real forests in a switching class are isomorphic. Here a forest is real, if it is disconnected.

Degree conditions for vertex switching reconstruction

We give bounds on the maximum and the minimum degree of a graph for s-vertex switching reconstruction problem.

Applications of balance equations to vertex switching reconstruction

AbstractA graph is called s‐vertex switching reconstructible (s‐VSR) if it is uniquely defined, up to isomorphism, by the multiset of unlabeled graphs obtained by switching of all its s‐vertex subsets. We show that a graph with n vertices is n/2‐VSR if n = 0(mod 4), (n − 2)/2‐VSR and n/2‐VSR if n = 2(mod 4), (n − 1)/2‐VSR if n = 1 (mod 2). For hypothetical nonreconstructible graphs, we give bounds on the number of edges (for any s) and on the maximum and minimum degree (for s = 2). We also show that for n > 9 the degree sequence is 2‐VSR.