Given a graph G=(V, E), where V and E are vertex and edge sets of G, and a subset VNT of vertices called a non-terminal set, a spanning tree with a non-terminal set VNT, denoted by STNT, is a connected and acyclic spanning subgraph of G that contains all vertices of V where each vertex in a non-terminal set is not a leaf. On general graphs, the problem of finding an STNT of G is known to be NP-hard. In this paper, we show that if G is a circular-arc graph then finding an STNT of G is polynomially solvable with respect to the number of vertices.

The minimum vertex ranking spanning tree problem is to find a spanning tree of G whose vertex ranking is minimum. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of interval graphs. This paper proposes a polynomial time algorithm for solving the minimum vertex ranking spanning tree problem on outerplanar graphs.

This paper analyzes what structural features of graph problems allow efficient parallel algorithms. We survey some parallel algorithms for typical problems on three kinds of graphs, outerplanar graphs, trapezoid graphs and in-tournament graphs. Our results on the shortest path problem, the longest path problem and the maximum flow problem on outerplanar graphs, the minimum-weight connected dominating set problem and the coloring problem on trapezoid graphs and Hamiltonian path and Hamiltonian cycle problem on in-tournament graphs are adopted as working examples.

Given a graph G=(V,E) where V and E are a vertex and an edge set, respectively, specified with a subset VNT of vertices called a non-terminal set, the spanning tree with non-terminal set VNT 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 VNT 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 non-terminal set VNT of G is linearly-solvable when each edge has the weight of one.

Given a graph G=(V, E), where V and E are vertex and edge sets of G, and a subset VNT of vertices called a non-terminal set, the minimum spanning tree with a non-terminal set VNT, denoted by MSTNT, is a connected and acyclic spanning subgraph of G that contains all vertices of V with the minimum weight where each vertex in a non-terminal set is not a leaf. On general graphs, the problem of finding an MSTNT of G is NP-hard. We show that if G is an outerplanar graph then finding an MSTNT of G is linearly solvable with respect to the number of vertices.

The minimum vertex ranking spanning tree problem is to find a spanning tree of G whose vertex ranking is minimum. This paper proposes an O(n3) time algorithm for solving the minimum vertex ranking spanning tree problem on an interval graph.

Given a graph G=(V,E), where V and E are vertex and edge sets of G, and a subset VNT of vertices called a non-terminal set, the minimum spanning tree with a non-terminal set VNT, denoted by MSTNT, is a connected and acyclic spanning subgraph of G that contains all vertices of V with the minimum weight where each vertex in a non-terminal set is not a leaf. On general graphs, the problem of finding an MSTNT of G is NP-hard. We show that if G is a series-parallel graph then finding an MSTNT of G is linearly solvable with respect to the number of vertices.

Given a graph G=(V,E) where V and E are a vertex and an edge set, respectively, specified with a subset VNT of vertices called a non-terminal set, the spanning tree with non-terminal set VNT 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. In the case where each edge has the weight of a nonnegative integer, the problem of finding a minimum spanning tree with a non-terminal set VNT of G was known to be NP-hard. However, the complexity of finding a spanning tree on general graphs where each edge has the weight of one was unknown. In this paper, we consider this problem and first show that it is NP-hard even if each edge has the weight of one on general graphs. We also show that if G is a cograph then finding a spanning tree with a non-terminal set VNT of G is linearly solvable when each edge has the weight of one.

As a super class of tournament digraphs, Bang-Jensen, Huang and Prisner defined an in-tournament digraph (in-tournament for short) and investigated a number of its nice properties. The in-tournament is a directed graph in which the set of in-neighbors of every vertex induces a tournament digraph. In other words, the presence of arcs (x,z) and (y,z) implies that exactly one of (x,y) or (y,x) exists. In this paper, we propose, for in-tournaments, parallel algorithms for examining the existence of a Hamiltonian path and a Hamiltonian cycle and for constructing them, if they exist.

This paper presents an O(n2)-time algorithm for constructing two edge-disjoint paths connecting two given pairs of vertices in a given tournament graph. It improves the time complexity of a previously known O(n4)-time algorithm.