Total Vertex Research Articles

On total vertex irregularity strength of graphs

On total vertex irregularity strength of graphs

Total domination critical graphs with maximal order

A graph G without isolated vertices is total domination vertex critical, if for any vertex v of G that is not adjacent to a pendant vertex, the total domination number of G-v is less than the total domination number of G. We call these graphs γt-critical. If G is a γt-critical graph of order n, then it can be shown that n≤△(G)(γt(G)-1)+1, where △(G) is the maximum degree of G. In this paper, we characterize the graphs in the case n≤△(G)(γt(G)-1)+1 with γt(G) = 3.

Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs

A vertex irregular total -labeling of a graph with vertex set and edge set is an assignment of positive integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of , denoted by is the minimum value of the largest label over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of isomorphic sun graphs, , disjoint union of consecutive nonisomorphic sun graphs, , and disjoint union of any two nonisomorphic sun graphs .

Irregular Total Labellings of Generalized Petersen Graphs

The total edge irregularity strength tes(G) and total vertex irregularity strength tvs(G) are invariants analogous to irregular strength s(G) of a graph G for total labellings. Baca et al. (Discrete Math. 307:1378–1388, 2007) determined the bounds and precise values for some families of graphs concerning these parameters. In this paper, we show the exact values of the total edge irregularity strength and total vertex irregularity strength of generalized Petersen graphs P(n,k).

Total domination changing and stable graphs upon vertex removal

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. A graph is total domination vertex removal stable if the removal of an arbitrary vertex leaves the total domination number unchanged. On the other hand, a graph is total domination vertex removal changing if the removal of an arbitrary vertex changes the total domination number. In this paper, we study total domination vertex removal changing and stable graphs.

On the existence problem of the total domination vertex critical graphs

The existence problem of the total domination vertex critical graphs has been studied in a series of articles. We first settle the existence problem with respect to the parities of the total domination number m and the maximum degree Δ : for even m except m = 4 , there is no m - γ t -critical graph regardless of the parity of Δ ; for m = 4 or odd m ≥ 3 and for even Δ , an m - γ t -critical graph exists if and only if Δ ≥ 2 ⌊ m − 1 2 ⌋ ; for m = 4 or odd m ≥ 3 and for odd Δ , if Δ ≥ 2 ⌊ m − 1 2 ⌋ + 7 , then m - γ t -critical graphs exist, if Δ < 2 ⌊ m − 1 2 ⌋ , then m - γ t -critical graphs do not exist. The only remaining open cases are Δ = 2 ⌊ m − 1 2 ⌋ + k , k = 1 , 3 , 5 . Second, we study these remaining open cases when m = 4 or odd m ≥ 9 . As the previously known result for m = 3 , we also show that for Δ ( G ) = 3 , 5 , 7 , there is no 4 - γ t -critical graph of order Δ ( G ) + 4 . On the contrary, it is shown that for odd m ≥ 9 there exists an m - γ t -critical graph for all Δ ≥ m − 1 .

On the total vertex irregularity strength of trees

A vertex irregular total k -labelling λ : V ( G ) ∪ E ( G ) ⟶ { 1 , 2 , … , k } of a graph G is a labelling of vertices and edges of G done in such a way that for any different vertices x and y , their weights w t ( x ) and w t ( y ) are distinct. The weight w t ( x ) of a vertex x is the sum of the label of x and the labels of all edges incident with x . The minimum k for which a graph G has a vertex irregular total k -labelling is called the total vertex irregularity strength of G , denoted by tvs ( G ) . In this paper, we determine the total vertex irregularity strength of trees.

Matching Properties in Total Domination Vertex Critical Graphs

A vertex subset S of a graph G = (V,E) is a total dominating set if every vertex of G is adjacent to some vertex in S. The total domination number of G, denoted by γ t (G), is the minimum cardinality of a total dominating set of G. A graph G with no isolated vertex is said to be total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, γ t (G−v) < γ t (G). A total domination vertex critical graph G is called k-γ t -critical if γ t (G) = k. In this paper we first show that every 3-γ t -critical graph G of even order has a perfect matching if it is K 1,5-free. Secondly, we show that every 3-γ t -critical graph G of odd order is factor-critical if it is K 1,5-free. Finally, we show that G has a perfect matching if G is a K 1,4-free 4-γ t (G)-critical graph of even order and G is factor-critical if G is a K 1,4-free 4-γ t (G)-critical graph of odd order.

A new upper bound for the total vertex irregularity strength of graphs

We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting w of a graph G = ( V , E ) with elements of a set { 1 , 2 , … , s } , denote w t G ( v ) = ∑ e ∋ v w ( e ) + w ( v ) for each v ∈ V . The smallest s for which exists such a weighting with w t G ( u ) ≠ w t G ( v ) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of this graph, and is denoted by tvs ( G ) . We prove that tvs ( G ) ≤ 3 ⌈ n / δ ⌉ + 1 for each graph of order n and with minimum degree δ > 0 .

On total domination vertex critical graphs of high connectivity

A graph is called γ -critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γ t -critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γ t -critical graph that has total domination number k , is called k - γ t -critical. In this paper, we introduce a class of k - γ t -critical graphs of high connectivity for each integer k ≥ 3 . In particular, we provide a partial answer to the question “Which graphs are γ -critical and γ t -critical or one but not the other?” posed in a recent work [W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255–261].

Linear Bound on the Irregularity Strength and the Total Vertex Irregularity Strength of Graphs

Let G be a simple graph of order n with no isolated edges and at most one isolated vertex. For a positive integer w, a w-weighting of G is a function $f:E(G)\rightarrow\{1,2,\dots,w\}$. An irregularity strength of G, $s(G)$, is the smallest w such that there is a w-weighting of G for which $\sum_{e:u\in e}f(e)\neq\sum_{e:v\in e}f(e)$ for all pairs of different vertices $u,v\in V(G)$. We prove that $s(G)<112\frac{n}{\delta}+28$, where $\delta$ is the minimum degree of G. For d-regular graphs, we strengthen this to $s(G)<40\frac{n}{d}+11$. These upper bounds represent improvements of many existing ones. Similar results concerning the “total” version of the irregularity strength are also discussed.

Chain partitioning as a key element for building vehicle routing problem heuristics

Chain partitioning is the process of partitioning a tree into chains. The edges and vertices of the tree have weights. In a vehicle routing context the weights of edges might represent distances between customers and the weights of the vertices might represent the demand of the customers. Based on the distances a minimum spanning tree can be built and, when partitioned into chains with total vertex weight per chain less than a certain value, each chain represents a route with total demand less than the vehicle capacity. This one -step process can be embedded in a local search procedure or a meta-heuristic with the advantage that a neighbour always represents a feasible solution

A constructive characterization of total domination vertex critical graphs

Let G be a graph of order n and maximum degree Δ ( G ) and let γ t ( G ) denote the minimum cardinality of a total dominating set of a graph G . A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G − v is less than the total domination number of G . We call these graphs γ t -critical. For any γ t -critical graph G , it can be shown that n ≤ Δ ( G ) ( γ t ( G ) − 1 ) + 1 . In this paper, we prove that: Let G be a connected γ t -critical graph of order n ( n ≥ 3 ), then n = Δ ( G ) ( γ t ( G ) − 1 ) + 1 if and only if G is regular and, for each v ∈ V ( G ) , there is an A ⊆ V ( G ) − { v } such that N ( v ) ∩ A = 0̸ , the subgraph induced by A is 1-regular, and every vertex in V ( G ) − A − { v } has exactly one neighbor in A .

On irregular total labellings

Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved.

On an open problem concerning total domination critical graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G. We call these graphs γ t -critical. If such a graph G has total domination number k, we call it k- γ t -critical. We verify an open problem of k- γ t -critical graphs and obtain some results on the characterization of total domination critical graphs of order n = Δ ( G ) ( γ t ( G ) - 1 ) + 1 .

On the total domination critical graphs

Abstract A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G ∖ { v } is less than the total domination number of G. These graphs we call γ t -critical. If such a graph G has total domination number k, we call it k- γ t -critical. We verify an open problem of k- γ t -critical graphs and obtain some results on the characterization of total domination critical graphs of order Δ ( G ) + γ t ( G ) .

LMI conditions for quadratic and robust -stability of interval systems

Sufficient conditions for the quadratic Open image in new window -stability and further robust Open image in new window -stability of interval systems are presented in this paper. This robust Open image in new window -stability condition is based on a parameter-dependent Lyapunov function obtained from the feasibility of a set of linear matrix inequalities (LMIs) defined at a series of partial-vertex-based interval matrices other than the total vertex matrices as in previous results. The results contain the usual quadratic and robust stability of continuous-time and discrete-time interval systems as particular cases. The illustrative example shows that this method is effective and less conservative for checking the quadratic and robust Open image in new window -stability of interval systems.

The diameter of total domination vertex critical graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G. These graphs we call γ t -critical. If such a graph G has total domination number k, we call it k- γ t -critical. We characterize the connected graphs with minimum degree one that are γ t -critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k- γ t -critical graph for k ⩽ 8 and provide an example which shows that the maximum diameter is in general at least 5 k / 3 - O ( 1 ) .

Metallaboranes of the earlier transition metals. An arachno nine-vertex, nine-skeletal electron pair rhenaborane of novel shape: importance of total vertex connectivities in such systems

The synthesis and characterization of Cp*2Re2B7H11 provides the first example of an open hypoelectronic metallaborane which is shown to possess an arachno geometry based on a total vertex connectivity identical with that of a canonical arachno nine-vertex, twelve-skeletal electron pair cluster.

Total Colourings of Planar Graphs with Large Girth

It is proved that ifGis a planar graph with total (vertex–edge) chromatic number χ″, maximum degree Δ and girthg, then χ″=Δ+1 if Δ≥5 andg≥5, or Δ≥4 andg≥6, or Δ≥3 andg≥10. These results hold also for graphs in the projective plane, torus and Klein bottle.