ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β a... more ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in α,β-set V'. Graphs which are not α,β destructible for any α,β are called stable, If G is a stable graph on a prime number p ≥ 7 of vertices, then we show that G has a maximum number of edges if and only if G is K2,p-2, We also characterize stable graphs on a minimum number of edges.
Let G = (V, E) be a graph of order n � 2. The double vertex graph, U2(G), is the graph whose vert... more Let G = (V, E) be a graph of order n � 2. The double vertex graph, U2(G), is the graph whose vertex set consists of all n 2 � un- ordered pairs from V such that two vertices {x, y} and {u, v} are adjacent if and only if |{x, y} T {u, v}| = 1 and if x
A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are integr... more A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are integral factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in a β-set V'. Graphs which are not α, β destructible for any α.β are called stable. In this paper we prove that all
ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β a... more ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in α,β-set V'. Graphs which are not α,β destructible for any α,β are called stable, If G is a stable graph on a prime number p ≥ 7 of vertices, then we show that G has a maximum number of edges if and only if G is K2,p-2, We also characterize stable graphs on a minimum number of edges.
International Journal of Foundations of Computer Science, 2012
ABSTRACT Self-stabilizing algorithms represent an extension of distributed algorithms in which no... more ABSTRACT Self-stabilizing algorithms represent an extension of distributed algorithms in which nodes of the network have neither coordination, synchronization, nor initialization. We consider the model where there is one designated master node and all other nodes are anonymous and have constant space. Recently, Lee et al. obtained such an algorithm for determining the size of a unidirectional ring. We provide a new algorithm that converges much quicker. This algorithm exploits a token-circulation idea due to Afek and Brown. Disregarding the time for stabilization, our algorithm computes the size of the ring at the master node in O(n log n) time compared to O(n3) steps used in the algorithm by Lee et al. We have also shown that the master node, after determining the size of the ring, can compute the average of observations made at each node in O(n) rounds or O(n2) steps. It seems likely that one should be able to obtain master–slave algorithms for other problems in networks.
A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus ... more A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus γ(G) is obtained. For γ > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G) > ϰ(G)/2 − 1. For ϰ = 1 this bound is
ABSTRACT Let G=(V,E) be a graph, S⊆V. A vertex a is called a dominator of S if it is adjacent to ... more ABSTRACT Let G=(V,E) be a graph, S⊆V. A vertex a is called a dominator of S if it is adjacent to every vertex w∈S. The domination number γ(G) of a graph G is the minimum order of a dominating set of G. A dominator partition of a graph G is a partition Π={V 1 ,⋯,V l } of V such that every vertex v∈V is a dominator of at least one class V i ∈Π. A dominator partition Π={V 1 ,⋯,V l } is minimal, if any partition Π ' obtained from Π by forming the union of any two classes V i ,V j , i≠j, as one class of Π ' is no longer a dominator partition. The dominator partition number of a graph G, denoted π d (G), is the minimum order of a dominant partition of G. The upper dominator partition number of a graph G, denoted Π d (G), is the maximum order of a minimal dominator partition of G. It is known that n/(1+Δ(G))≤π d (G)≤Π d (G)≤n-δ(G). In this paper it is proved that for any tree T on n vertices, n≥4, Π d (G)>n/2 (Theorem 3.5); Π d (P n )≥[(2n+1)/3] (Theorem 4.3) and Π d (C n )≥[(2n+1)/3] for n≥5 (Theorem 4.9).
ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β a... more ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in α,β-set V'. Graphs which are not α,β destructible for any α,β are called stable, If G is a stable graph on a prime number p ≥ 7 of vertices, then we show that G has a maximum number of edges if and only if G is K2,p-2, We also characterize stable graphs on a minimum number of edges.
Let G = (V, E) be a graph of order n � 2. The double vertex graph, U2(G), is the graph whose vert... more Let G = (V, E) be a graph of order n � 2. The double vertex graph, U2(G), is the graph whose vertex set consists of all n 2 � un- ordered pairs from V such that two vertices {x, y} and {u, v} are adjacent if and only if |{x, y} T {u, v}| = 1 and if x
A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are integr... more A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are integral factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in a β-set V'. Graphs which are not α, β destructible for any α.β are called stable. In this paper we prove that all
ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β a... more ABSTRACT A connected, nontrivial, simple graph of order v is said to be α,β destructible if α,β are factors of v and an α-set of edges, E', exists whose removal from G isolates exactly the vertices in α,β-set V'. Graphs which are not α,β destructible for any α,β are called stable, If G is a stable graph on a prime number p ≥ 7 of vertices, then we show that G has a maximum number of edges if and only if G is K2,p-2, We also characterize stable graphs on a minimum number of edges.
International Journal of Foundations of Computer Science, 2012
ABSTRACT Self-stabilizing algorithms represent an extension of distributed algorithms in which no... more ABSTRACT Self-stabilizing algorithms represent an extension of distributed algorithms in which nodes of the network have neither coordination, synchronization, nor initialization. We consider the model where there is one designated master node and all other nodes are anonymous and have constant space. Recently, Lee et al. obtained such an algorithm for determining the size of a unidirectional ring. We provide a new algorithm that converges much quicker. This algorithm exploits a token-circulation idea due to Afek and Brown. Disregarding the time for stabilization, our algorithm computes the size of the ring at the master node in O(n log n) time compared to O(n3) steps used in the algorithm by Lee et al. We have also shown that the master node, after determining the size of the ring, can compute the average of observations made at each node in O(n) rounds or O(n2) steps. It seems likely that one should be able to obtain master–slave algorithms for other problems in networks.
A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus ... more A new lower bound on the toughness t(G) of a graph G in terms of its connectivity ϰ(G) and genus γ(G) is obtained. For γ > 0, the bound is sharp via an infinite class of extremal graphs all of girth 4. For planar graphs, the bound is t(G) > ϰ(G)/2 − 1. For ϰ = 1 this bound is
ABSTRACT Let G=(V,E) be a graph, S⊆V. A vertex a is called a dominator of S if it is adjacent to ... more ABSTRACT Let G=(V,E) be a graph, S⊆V. A vertex a is called a dominator of S if it is adjacent to every vertex w∈S. The domination number γ(G) of a graph G is the minimum order of a dominating set of G. A dominator partition of a graph G is a partition Π={V 1 ,⋯,V l } of V such that every vertex v∈V is a dominator of at least one class V i ∈Π. A dominator partition Π={V 1 ,⋯,V l } is minimal, if any partition Π ' obtained from Π by forming the union of any two classes V i ,V j , i≠j, as one class of Π ' is no longer a dominator partition. The dominator partition number of a graph G, denoted π d (G), is the minimum order of a dominant partition of G. The upper dominator partition number of a graph G, denoted Π d (G), is the maximum order of a minimal dominator partition of G. It is known that n/(1+Δ(G))≤π d (G)≤Π d (G)≤n-δ(G). In this paper it is proved that for any tree T on n vertices, n≥4, Π d (G)>n/2 (Theorem 3.5); Π d (P n )≥[(2n+1)/3] (Theorem 4.3) and Π d (C n )≥[(2n+1)/3] for n≥5 (Theorem 4.9).
Uploads
Papers by Wayne Goddard