Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Algorithms for VLSI Physcial Design Automation
Algorithms for VLSI Physcial Design Automation
Graph Theory
| Hi-index | 0.00 |
Let G be a nontrivial connected graph of order n and let k be an integer with 2驴k驴n. For a set S of k vertices of G, let 驴(S) denote the maximum number 驴 of edge-disjoint trees T 1,T 2,驴,T 驴 in G such that V(T i )驴V(T j )=S for every pair i,j of distinct integers with 1驴i,j驴驴. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by 驴 k (G), of G is defined by 驴 k (G)=min{驴(S)}, where the minimum is taken over all k-subsets S of V(G). Thus 驴 2(G)=驴(G), where 驴(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it.This paper mainly focus on the complexity of determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k 1 and k 2, given a graph G and a k 1-subset S of V(G), the problem of deciding whether G contains k 2 internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k 1 is a fixed integer of at least 4, but k 2 is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k 2 is a fixed integer of at least 2, but k 1 is not a fixed integer, we show that the problem also becomes NP-complete.