A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken ove...A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.展开更多
A. Kaneko and K. Ota proved that for a minimally （n, λ）-connected graph G, if ｜G｜ = p ≥ 3n - 1, then e（G） ≤ nλ（｜G｜ - n）; and if e（G） = nλ（｜G｜ - n）, then G is isomorphic to the graph Kn^λ,p-n whic...A. Kaneko and K. Ota proved that for a minimally （n, λ）-connected graph G, if ｜G｜ = p ≥ 3n - 1, then e（G） ≤ nλ（｜G｜ - n）; and if e（G） = nλ（｜G｜ - n）, then G is isomorphic to the graph Kn^λ,p-n which is obtained from the complete bipartite graph Kn,p-n by replacing each edge with A multiple edges; if 3n - 1 ≥ ｜G}≥ n ＋ 1, then e（G） ≤λ（｜G｜ ＋ n）^2/8. In this paper, we determine all the minimally （n, λ）-connected graphs with order p and the maximum size λ（p＋n）^2/8 for 3n- 1 ≥ p ≥ n＋ 1 for 3n-1≥p≥n＋1.展开更多
M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + ...M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + 2, n} or G belongs to one exceptional class of graphs.展开更多
It is difficult to characterize graphs which contain no a 2-connected graph as a minor in graph theory.Let V_(8)be a graph constructed from an 8-cycle by connecting the antipodal vertices.There are thirteen 2-connecte...It is difficult to characterize graphs which contain no a 2-connected graph as a minor in graph theory.Let V_(8)be a graph constructed from an 8-cycle by connecting the antipodal vertices.There are thirteen 2-connected spanning subgraphs of V_(8).In particular,one of them is obtained from the Petersen graph by deleting two vertices and it is also a hard problem to characterize Petersen-minor-free graphs.In this paper,we characterize internally 4-connected graphs which contain a 2-connected spanning subgraph of V_(8)as a forbidden minor.展开更多
A connected graph G is said to be a factor-critical graph if G - v has a perfect matching for every vertex v of G. In this paper, the 2-connected factor-critical graph G which has exactly ｜E（G）｜＋ 1 maximum matchi...A connected graph G is said to be a factor-critical graph if G - v has a perfect matching for every vertex v of G. In this paper, the 2-connected factor-critical graph G which has exactly ｜E（G）｜＋ 1 maximum matchings is characterized.展开更多
Let B（G） denote the bipartite double cover of a non-bipartite graph G with v ≥ 2 vertices and s edges. We prove that G is a perfect 2-matching covered graph if and only if B（G） is a 1-extendable graph. Furthermor...Let B（G） denote the bipartite double cover of a non-bipartite graph G with v ≥ 2 vertices and s edges. We prove that G is a perfect 2-matching covered graph if and only if B（G） is a 1-extendable graph. Furthermore, we prove that B（G） is a minimally l-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e = xy ∈ E（G）, there is an independent set S in G such that ｜ГG（S）｜ = ｜S｜ ＋ 1, x ∈ S and ｜ГG-xy（S）｜ = ｜S｜ Then, we construct a digraph D from B（G） or G and show that D is a strongly connected digraph if and only if G is a perfect 2-matching covered graph. So we design an algorithm in O（x√vε） time that determines whether G is a perfect 2-matching covered graph or not.展开更多
The directed L-distance minimal dominating set(MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theo...The directed L-distance minimal dominating set(MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theoretical interest. We only give results for an Erdós Rényi(ER)random graph and regular random(RR) graph, but this work can be extended to any type of network. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation(BP) algorithm does not converge when the inverse temperatureβ exceeds a threshold on either an ER random network or RR network. Second, the entropy density of replica symmetric theory has a transition point at a finite β on a regular random graph when the arc density exceeds 2 and on an ER random graph when the arc density exceeds3.3;there is no entropy transition point(or β = ■) in other circumstances. Third, the results of the replica symmetry(RS) theory are in agreement with those of BP algorithm while the results of the BP decimation algorithm are better than those of the greedy heuristic algorithm.展开更多
We discuss k-factors and Hamiltonian Graphs in graph theory. We prove a general version of the conjecture by R. Haggkvist; as a result, we prove two extended versions of two well-known theorems due to O. Ore and B. Ja...We discuss k-factors and Hamiltonian Graphs in graph theory. We prove a general version of the conjecture by R. Haggkvist; as a result, we prove two extended versions of two well-known theorems due to O. Ore and B. Jachson, respectively.展开更多
Many results have been obtained in investigating the existence of Hamiltonian cycles in 2-connected, k-regular graphs, see [3], [1], [4], [6] and [2].We consider only simple graphs here and use standard graph theory n...Many results have been obtained in investigating the existence of Hamiltonian cycles in 2-connected, k-regular graphs, see [3], [1], [4], [6] and [2].We consider only simple graphs here and use standard graph theory notations and terminology. We let V(G) and E(G) denote the vertex set and the edge set of graph G respectively.展开更多
In[1],P.Paulraja posed the following problem:Let G be a 2-connected graph suchthat δ(G)≥3,where δ(G)denotes the minimum degree of G.If each edge of G lies on either a cycle oflength 3 or a cycle of length 4,is it t...In[1],P.Paulraja posed the following problem:Let G be a 2-connected graph suchthat δ(G)≥3,where δ(G)denotes the minimum degree of G.If each edge of G lies on either a cycle oflength 3 or a cycle of length 4,is it true that G has a spanning Eulerian subgraph?A related case inwhich δ(G)≥4 is settled affairmatively in this paper.展开更多
文摘A subset S of V is called a k-connected dominating set if S is a dominating set and the induced subgraph S has at most k components.The k-connected domination number γck（G） of G is the minimum cardinality taken over all minimal k-connected dominating sets of G.In this paper,we characterize trees and unicyclic graphs with equal connected domination and 2-connected domination numbers.
基金This research is supported by the National Natural Science Foundation of China.
文摘A. Kaneko and K. Ota proved that for a minimally （n, λ）-connected graph G, if ｜G｜ = p ≥ 3n - 1, then e（G） ≤ nλ（｜G｜ - n）; and if e（G） = nλ（｜G｜ - n）, then G is isomorphic to the graph Kn^λ,p-n which is obtained from the complete bipartite graph Kn,p-n by replacing each edge with A multiple edges; if 3n - 1 ≥ ｜G}≥ n ＋ 1, then e（G） ≤λ（｜G｜ ＋ n）^2/8. In this paper, we determine all the minimally （n, λ）-connected graphs with order p and the maximum size λ（p＋n）^2/8 for 3n- 1 ≥ p ≥ n＋ 1 for 3n-1≥p≥n＋1.
文摘M. Matthews and D. Sumner proved that if G is a 2-connected claw-free graph of order n, then c(G) min{2δb + 4, n}. In this paper, we prove that if G is a,2-connected claw-free graph on n venices, then c(G) min{3δ + 2, n} or G belongs to one exceptional class of graphs.
基金Research Project supported by the National Natural Science Foundation of China(No.11961051)Provincial Natural Science Foundation of Shanxi(20210302123097)Shanxi Scholarship Council of China。
文摘It is difficult to characterize graphs which contain no a 2-connected graph as a minor in graph theory.Let V_(8)be a graph constructed from an 8-cycle by connecting the antipodal vertices.There are thirteen 2-connected spanning subgraphs of V_(8).In particular,one of them is obtained from the Petersen graph by deleting two vertices and it is also a hard problem to characterize Petersen-minor-free graphs.In this paper,we characterize internally 4-connected graphs which contain a 2-connected spanning subgraph of V_(8)as a forbidden minor.
基金supported by the National Natural Science Foundation of China(No.11551003)the Scientific research fund of the Science and Technology Program of Guangzhou,China(No.201510010265)the Qinghai Province Natural Science Foundation(No.2015-ZJ-911)
文摘A connected graph G is said to be a factor-critical graph if G - v has a perfect matching for every vertex v of G. In this paper, the 2-connected factor-critical graph G which has exactly ｜E（G）｜＋ 1 maximum matchings is characterized.
基金Acknowledgements The authors would like to thank the referees very much for their careful reading and their very instructive suggestions which improve this paper greatly. This work was supported in part by the National Natural Science Foundation of China （Grant No. 11201158）.
文摘Let B（G） denote the bipartite double cover of a non-bipartite graph G with v ≥ 2 vertices and s edges. We prove that G is a perfect 2-matching covered graph if and only if B（G） is a 1-extendable graph. Furthermore, we prove that B（G） is a minimally l-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e = xy ∈ E（G）, there is an independent set S in G such that ｜ГG（S）｜ = ｜S｜ ＋ 1, x ∈ S and ｜ГG-xy（S）｜ = ｜S｜ Then, we construct a digraph D from B（G） or G and show that D is a strongly connected digraph if and only if G is a perfect 2-matching covered graph. So we design an algorithm in O（x√vε） time that determines whether G is a perfect 2-matching covered graph or not.
基金supported by the doctoral startup fund of Xinjiang University of China (grant number 208-61357)the National Natural Science Foundation of China (grant number 11 465 019,11 664 040)。
文摘The directed L-distance minimal dominating set(MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theoretical interest. We only give results for an Erdós Rényi(ER)random graph and regular random(RR) graph, but this work can be extended to any type of network. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation(BP) algorithm does not converge when the inverse temperatureβ exceeds a threshold on either an ER random network or RR network. Second, the entropy density of replica symmetric theory has a transition point at a finite β on a regular random graph when the arc density exceeds 2 and on an ER random graph when the arc density exceeds3.3;there is no entropy transition point(or β = ■) in other circumstances. Third, the results of the replica symmetry(RS) theory are in agreement with those of BP algorithm while the results of the BP decimation algorithm are better than those of the greedy heuristic algorithm.
基金supported by the Key Laboratory of Power System,Tsinghua University
文摘We discuss k-factors and Hamiltonian Graphs in graph theory. We prove a general version of the conjecture by R. Haggkvist; as a result, we prove two extended versions of two well-known theorems due to O. Ore and B. Jachson, respectively.
文摘Many results have been obtained in investigating the existence of Hamiltonian cycles in 2-connected, k-regular graphs, see [3], [1], [4], [6] and [2].We consider only simple graphs here and use standard graph theory notations and terminology. We let V(G) and E(G) denote the vertex set and the edge set of graph G respectively.
基金The present work is supported by the National Natural Science Foundation of China
文摘In[1],P.Paulraja posed the following problem:Let G be a 2-connected graph suchthat δ(G)≥3,where δ(G)denotes the minimum degree of G.If each edge of G lies on either a cycle oflength 3 or a cycle of length 4,is it true that G has a spanning Eulerian subgraph?A related case inwhich δ(G)≥4 is settled affairmatively in this paper.
