We consider the problem of placing arrows, which indicate the direction of each edge in directed graph drawings, without making them overlap other arrows, vertices and edges as much as possible. The following two methods have been proposed for this problem. One is an exact algorithm for the case in which the position of each arrow is restricted to some discrete points. The other is a heuristic algorithm for the case in which the arrow is allowed to move continuously on each edge. In this paper, we assume that the arrow positions are not restricted to discrete points and propose an exact algorithm for the problem of finding an arrow placement such that (a) the weighted sum of the numbers of overlaps with edges, vertices and other arrows is minimized and (b) the sum of the distances between the arrows and their edges' terminal vertices is minimized as a secondary objective. The proposed method solves this problem by reducing it to a mixed integer linear programming problem. Since this is an exponential time algorithm, we add a simple procedure as preprocessing to reduce the running time. Experimental results show that the proposed method can find a better arrow placement than the previous methods and the procedure for reducing the running time is effective.

A graph is said to be outerplanar if it has a planar embedding in which all vertices lie on the exterior face. For a graph G=(V, E), its spanning subgraph G =(V, E ) is called a maximal outerplanar spanning subgraph (abbreviated as mosg) if G is outerplanar and (V, E") is not outerplanar for any edge set E" such that E E"E. In this paper, we present a linear-time algorithm for finding an mosg of a given graph.

Suppose that there are terminals on two concentric circles Cin and Cout, with Cin inside of Cout. A set of two-terminal nets is given and the routing area is the annular region between the two circles. In this paper, we present an O(n2) time algorithm for testing whether the given net set is two-layer routable, where n is the number of nets. Applying this algorithm repeatedly, we can find, in O(n3) time, a maximal subset of nets which is two-layer routable.

Suppose that there are terminals on two concentric circles, Cin and Cout, with Cin inside of Cout. We are given a set of nets each of which consists of a terminal on Cin and a terminal on Cout. The routing area is the annular region between the two circles. In this paper, we present an O(nk-1) time algorithm for testing whether the given net set is k-layer routable without vias, where k2 and n is the number of nets.

Several two-dimensional largest common subpatterns (LCP) between pictures are defined and their computing methods are proposed. The time and space complexities of the computing methods are O(IJMN) to obtain the size of LCPs between a picture with IJ pixels and a picture with MN pixels. These LCPs can be used as similarity measures between pictures and can be applied to texture recognition and classification.

Many metrics between trees have been proposed. However, there is no research on a graph metric that can be applied to molecular graphs. And most of the reports on tree metrics have dealt with rooted and ordered trees. As the first step defining a graph metric for molecular graphs, this paper proposes a tree metric between unrooted and unordered trees. This metric is based on a mapping between trees that determines a transformation from one tree to another. The metric is the minimum weight among the weights of all possible transformations. The characteristics of the mapping are investigated. A top-down computing method is proposed using the characteristics of the mapping. The time and space complexities are OT(N 2aN 2b(N 3aN 3b)) and Os(N 2aN 2b), respectively, where Na and Nb are the numbers of vertices of the two trees. If the degrees of all vertices of the trees are bounded by a constant, the time complexity of the method is O (N 3aN 3b). The computing time to obtain the distance between a pair of molecular graphs using a computer (SUN SparcStation ELC) is 0.51 seconds on average for all the pairs of 111 molecular graphs that have 12.0 atoms on average. This methic can be applied to the clustering of molecular graphs.

Several distances between trees have been proposed. However, most of the reports on distances have dealt with rooted and ordered trees. This paper proposes two distances between unrooted and cyclically ordered trees (CO-trees) and their computing methods. A CO-tree is a tree embedded in a plane. These distances are defined based on Tai's mapping (TM) and a strongly structure preserving mapping (SSPM) between CO-trees. The time complexities to compute the distances between two CO-trees Ta and Tb are OT (N 2aN 2b) for the distance based on a TM and OT(mambNaNb) for that on an SSPM, respectively, where ma(mb) and Na(Nb) are the largest degree of a vertex and the number of vertices of Ta(Tb), respectively. The space complexities of both methods are Os(NaNb). Those distances can be applied to the clustering of CO-trees.

Let G be an undirected graph and let Γ be its drawing on a plane. Each vertex in G has a label with a specified size. In this paper, we consider the problem of placing the maximum number of vertex labels in Γ in such a way that they do not overlap any vertices, edges or other labels. By refining several portions of the Kakoulis-Tollis algorithm for labeling graphical features, we present a heuristic algorithm for this problem. Experimental results show that our algorithm can place more labels than previous algorithms.

Let H=(V(H),E(H)) be a directed graph with distinguished vertices s and t. An st-path in H is a simple directed path starting from s and ending at t. Let (H) be defined as { SS is the set of vertices on an st-path in H (s and t are excluded)}. For an undirected graph G=(V(G),E(G)) with V(G) V(H)- { s,t }, if the family of maximal independent sets of G coincides with (H), we call H an MIS-diagram for G. In this paper, we provide a necessary and sufficient condition for a directed graph to be an MIS-diagram for an undirected graph. We also show that an undirected graph G has an MIS-diagram iff G is a cocomparability graph. Based on the proof of the latter result, we can construct an efficient algorithm for generating all maximal independent sets of a cocomparability graph.

In this paper we present polynomial time algorithms for the following three problems on a Hamilton graph with a prescribed Hamilton circuit: (1) Given a Hamilton graph G with a prescribed Hamilton circuit, find a maximal planar Hamilton subgraph of G, (2) Given a Hamilton graph G and a planar Hamilton subgraph H of G, find a maximal planar Hamilton subgraph of G that contains H, and (3) Given an edge-weighted Hamilton graph G=(V, E), find a planar Hamilton subgraph G=(V, E) of G such that the addition of an edge e E-Edestroys the planarity, unless we delete an edge from Ehaving no less weight than e. The time complexities of the algorithms are O(n+m), O(m3/2) and O(m5/3), respectively, where n(resp., m) is the number of vertices (resp., edges) of the Hamilton graph. These algorithms are applicable to the Topological Via Minimization problem.

The 2-terminal shortest path problem is to find a shortest path between two specified vertices in a given graph G. In this paper, we consider this problem in the following situation: G is given before the two vertices are specified so that some preprocessing is allowed to reduce the response time. We present a method for calculating lower bounds of the length of the shortest path for any pair of vertices. Experimental results show that the A* algorithm with our method performs much better than previous methods.