Given a graph G = (V, E) where V and E are a vertex and an edge set, respectively, specified with a subset V NT of vertices called a non-terminal set, the spanning tree with non-terminal set V NT is a connected and acyclic spanning subgraph of G that contains all the vertices of V where each vertex in a non-terminal set is not a leaf. The complexity of finding a spanning tree with non-terminal set V NT on general graphs where each edge has the weight of one is known to be NP-hard. In this paper, we show that if G is an interval graph then finding a spanning tree with a nonterminal set V NT of G is linearly-solvable when each edge has the weight of one. key words: spanning tree, interval graph, algorithm
Abstract. This paper discusses the complexity of packing k-chains (simple paths of length k) into an undirected graph; the chains packed must be either vertex-disjoint or edge-disjoint. Linear-time algorithms are given for both problems when the graph is a tree, and for the edge-disjoint packing problem when the graph is general and k = 2. The vertex-disjoint packing problem for general graphs is shown to be NP-complete even when the graph has maximum degree three and k = 2. Similarly the edge-disjoint packing problem is NP-complete even when the graph has maximum degree four and k = 3.
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