Abstract
Detecting vertex disjoint paths is one of the central issues in designing and evaluating an interconnection network. It is naturally related to routing among nodes and fault tolerance of the network. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and a path cover number of G denoted by p(G) = min{|P| : P is a path cover of G}. In this paper, we show that if the minimum degree sum of an independent set with k + 1 vertices in a connected quasi-claw-free graph G of order n is no less than n - k, then p(G) ≤ k - 1, where k ≥ 2. Examples illustrate that the degree sum condition in our result is sharp.
Highlights
It is well known that a multiprocessor system plays a crucial role in parallel and distributed computing. Such a system has an underlying topology, which is frequently represented as a graph in which vertices and edges correspond to nodes and links, respectively
Since a lot of mutually conflicting requirements are inevitable in designing the topology of an interconnection network, it is almost impossible to design a network which is optimum from all aspects
Finding parallel paths among nodes is one of the central issues concerned with efficient data transmission
Summary
It is well known that a multiprocessor system plays a crucial role in parallel and distributed computing. A path cover of a graph G is a spanning subgraph of G consisting of vertex disjoint paths, and let p(G) = min{|P| : P is a path cover of a graph G} denote the path cover number of G. The research on path cover number of a graph is a generalization of determining whether a graph is hamiltonian or traceable. It is natural to consider the same sufficient condition to determine whether a connected graph has path cover number less than k. If G is a connected quasi-claw-free graph of order n and σk+1(G) ≥ n − k, p(G) ≤ k − 1.
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