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.

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.

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.