Showing papers on "Path graph published in 1973"

Algorithm 447: efficient algorithms for graph manipulation

Abstract: Efficient algorithms are presented for partitioning a graph into connected components, biconnected components and simple paths. The algorithm for partitioning of a graph into simple paths of iterative and each iteration produces a new path between two vertices already on paths. (The start vertex can be specified dynamically.) If V is the number of vertices and E is the number of edges, each algorithm requires time and space proportional to max (V, E) when executed on a random access computer.

The Enumerative Theory of Planar Maps

Abstract: Publisher Summary This chapter discusses the enumerative theory of planar maps. A planar map is the figure formed when a non-null connected graph is drawn in the plane. The graph is allowed to have loops and multiple joins, and is required to be finite. The graph separates the rest of the plane into connected disjoint open regions called faces. Just one face is unbounded and is called the outer face. The vertices and edges of the defining graph G are called the vertices and edges, respectively, of the resulting planar map M, and they are assigned the same incidence relations with one another in M as in G. The frontier of any face is a union of edges and vertices of G. The face is said to be incident with those edges and vertices that lie in its frontier. The valency of a vertex of M is the number of incident edges, loops being counted twice.

On the existence of N‐connected graphs with prescribed degrees (n ≧ 2)

Abstract: In this paper we provide a simple algorithm for constructing an n- (vertex) connected graph with no multiple edges or loops having degree sequence satisfying the conditions indicated below. We prove these to be necessary and sufficient for the existence of such a graph. Our algorithm consists of constructing the graph in stages. An explicit construction is given if all degrees are n or n+1. Otherwise we choose a vertex with either the largest or smallest degree and connect it with a set of the remaining vertices having largest degrees, leaving a residual degree sequence for the rest of the graph which later can then be constructed by iteration. We connect the vertex of smallest degree above if the residual degrees after its connection are all at least n. Otherwise we connect the vertex of largest degree. The conditions on the degree sequence are that it must be realizable by a graph, all degrees must be at least n, and the number of edges which can avoid touching the n-1 largest degree vertices are sufficient to form a tree among the other vertices.

The existence of 1-factors in line graphs, squares, and total graphs

Abstract: The line graph L ( G ) of G is that graph whose vertex set corresponds to the edge set of G such that two vertices of L ( G ) are adjacent if and only if the corresponding edges of G are adjacent. The square G 2 of G has the same vertex set as G and two vertices are adjacent if and only if their distance in G is at most two. The total graph T ( G ) of G is the graph whose vertex set corresponds to the set of vertices and edges of G such that two vertices of T ( G ) are adjacent if and only if the corresponding elements of G are adjacent or incident. A 1-factor of a graph is a spanning 1-regular subgraph. For a connected graph G , it is shown that a necessary and sufficient condition that L ( G ) (respectively, G 2 ; respectively, T ( G )) have a 1-factor is that L ( G ) (respectively, G 2 ; respectively, T ( G )) have an even number of vertices.

Graphs with given connectivity and independence number or networks with given measures of vulnerability and survivability

Abstract: The criterion for invulnerability of a network based on the connectivity of a graph is well treated in literature. We define a criterion for survivability of a network in terms of the independence number of a graph. The following problems are then considered. 1) Find an r -connected graph, with n vertices and m_0 = [(nr + l)/2] edges, whose independence number is the minimum possible, where [x] denotes the greatest integer less than or equal to x \cdot 2 ) Given positive integers n,m,r , and k , find the realizability conditions for a graph with n vertices and m edges, whose connectivity is at least r and independence number is at most k .

Balanced Hypergraphs and Some Applications to Graph Theory

Abstract: Publisher Summary This chapter provides an overview of the combinatorial properties of a special kind of hypergraphs, called the balanced hypergraphs, which provide new theorems of graph theory. A hypergraph consists of a finite set X of n vertices together with a family of m nonempty subsets of X, called the edges. A graph is made up of vertices or nodes and lines called edges that connect them. A graph can be undirected, that is, there is no distinction between the two vertices associated with each edge, or its edges can be directed from one vertex to another; Graphs are one of the prime objects of study in discrete mathematics.

An Algorithm for Coloring the Vertices of an Arbitrary Finite Graph

Abstract: Let be G = (V,E) an arbitrary finite graph, where E is the set of edges and V= {vk}k=1n the set of n vertices numbered from 1 to n in an arbitrary order. We write (vk,v1) = ek1∈E if the vertices vk,v1∈V are connected by an edge otherwise ek1∉E.