A vertex subversion strategy of a graph G=(V,E) is a set of vertices S V(G) whose closed neighborhood is deleted from G . The survival subgraph is denoted by G/S . We call S a cut-strategy of G if G/S is disconnected,...A vertex subversion strategy of a graph G=(V,E) is a set of vertices S V(G) whose closed neighborhood is deleted from G . The survival subgraph is denoted by G/S . We call S a cut-strategy of G if G/S is disconnected, or is a clique, or is φ . The vertex-neighbor scattering number of G is defined to be VNS(G)=max{ω(G/S)-|S|} , where S is any cut-strategy of G , and ω(G/G) is the number of the components of G/S . It has been proved that the computing problem of this parameter is NP–complete, so we discuss the properties of vertex-neighbor-scattering number of trees in this paper.展开更多
It has been known that determining the exact value of vertex distinguishing edge index X ＇8（G） of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete...It has been known that determining the exact value of vertex distinguishing edge index X ＇8（G） of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let rid（G） denote the number of vertices of degree d in G, and let X＇es（G） be the equitable vertex distinguishing edge index of G. We show that a tree T holds nl （T） ≤ X ＇s （T） ≤ n1 （T） ＋ 1 and X＇s（T） = X＇es（T） if T satisfies one of the following conditions （i） n2（T） ≤△（T） or （ii） there exists a constant c with respect to 0 〈 c 〈 1 such that n2（T） △ cn1（T） and ∑3 ≤d≤△（T）nd（T） ≤ （1 - c）n1（T） ＋ 1.展开更多
The present paper shows the coordinates of a tree and its vertic es, defines a kind of Trees with Odd-Number Radiant Type (TONRT), deals with th e gracefulness of TONRT by using the edge-moving theorem, and uses gra...The present paper shows the coordinates of a tree and its vertic es, defines a kind of Trees with Odd-Number Radiant Type (TONRT), deals with th e gracefulness of TONRT by using the edge-moving theorem, and uses graceful TON RT to construct another class of graceful trees.展开更多
Many difficult (often NP-complete) optimization problems can be solved efficiently on graphs of small tree-width with a given tree-decomposition.In this paper,it is discussed how to solve the minimum feedback vertex s...Many difficult (often NP-complete) optimization problems can be solved efficiently on graphs of small tree-width with a given tree-decomposition.In this paper,it is discussed how to solve the minimum feedback vertex set problem and the minimum vertex feedback edge set problem efficiently by using dynamic programming on a tree-decomposition.展开更多
文摘A vertex subversion strategy of a graph G=(V,E) is a set of vertices S V(G) whose closed neighborhood is deleted from G . The survival subgraph is denoted by G/S . We call S a cut-strategy of G if G/S is disconnected, or is a clique, or is φ . The vertex-neighbor scattering number of G is defined to be VNS(G)=max{ω(G/S)-|S|} , where S is any cut-strategy of G , and ω(G/G) is the number of the components of G/S . It has been proved that the computing problem of this parameter is NP–complete, so we discuss the properties of vertex-neighbor-scattering number of trees in this paper.
基金supported by the National Natural Science Foundation of China (61163054),supported by the National Natural Science Foundation of China (61163037)
文摘It has been known that determining the exact value of vertex distinguishing edge index X ＇8（G） of a graph G is difficult, even for simple classes of graphs such as paths, cycles, bipartite complete graphs, complete, graphs, and graphs with maximum degree 2. Let rid（G） denote the number of vertices of degree d in G, and let X＇es（G） be the equitable vertex distinguishing edge index of G. We show that a tree T holds nl （T） ≤ X ＇s （T） ≤ n1 （T） ＋ 1 and X＇s（T） = X＇es（T） if T satisfies one of the following conditions （i） n2（T） ≤△（T） or （ii） there exists a constant c with respect to 0 〈 c 〈 1 such that n2（T） △ cn1（T） and ∑3 ≤d≤△（T）nd（T） ≤ （1 - c）n1（T） ＋ 1.
文摘The present paper shows the coordinates of a tree and its vertic es, defines a kind of Trees with Odd-Number Radiant Type (TONRT), deals with th e gracefulness of TONRT by using the edge-moving theorem, and uses graceful TON RT to construct another class of graceful trees.
基金Partially supported by the National Natural Science Foundation of China( 1 0 2 71 0 65
文摘Many difficult (often NP-complete) optimization problems can be solved efficiently on graphs of small tree-width with a given tree-decomposition.In this paper,it is discussed how to solve the minimum feedback vertex set problem and the minimum vertex feedback edge set problem efficiently by using dynamic programming on a tree-decomposition.
