A constructive characterization of contraction critical 8-connected graphs with minimum degree 9

2019 ◽  
Vol 342 (11) ◽  
pp. 3047-3056
Author(s):  
Chengfu Qin ◽  
Weihua He ◽  
Kiyoshi Ando
Keyword(s):  
Minimum Degree ◽  
Connected Graphs

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals On Extremal Graphs With No Long Paths

10.37236/1244 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Asad Ali Ali ◽  
William Staton
Keyword(s):  
Minimum Degree ◽  
Extremal Graphs ◽  
Connected Graphs ◽  
Long Paths

Connected graphs with minimum degree $\delta$ and at least $2\delta + 1$ vertices have paths with at least $2\delta + 1$ vertices. We provide a characterization of all such graphs which have no longer paths.

scholarly journals A recursive characterization of the 4-connected graphs

1990 ◽  
Vol 84 (1) ◽  
pp. 105-108 ◽  
Author(s):  
Nicola Martinov
Keyword(s):  
Connected Graphs

On the maximum connective eccentricity index among k-connected graphs

2020 ◽  
pp. 2150002
Author(s):  
Fazal Hayat
Keyword(s):  
Minimum Degree ◽  
Independence Number ◽  
Connected Graphs ◽  
Eccentricity Index

The connective eccentricity index (CEI for short) of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] and [Formula: see text] is the eccentricity of [Formula: see text] in [Formula: see text]. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the [Formula: see text]-vertex graphs with fixed connectivity and diameter, the [Formula: see text]-vertex graphs with fixed connectivity and independence number, and the [Formula: see text]-vertex graphs with fixed connectivity and minimum degree.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

scholarly journals Almost well-dominated bipartite graphs with minimum degree at least two

2020 ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Upper Domination Number ◽  
The Difference ◽  
Two Parameters

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

1980 ◽  
Vol 32 (6) ◽  
pp. 1325-1332 ◽  
Author(s):  
J. A. Bondy ◽  
R. C. Entringer
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Simple Graph ◽  
General Context ◽  
Connected Graphs ◽  
Longest Cycles ◽  
Image Position ◽  
The Relationship

The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

Sign in / Sign up

Export Citation Format

Share Document