In this paper, we investigate computational complexity of pipe puzzles. A pipe puzzle is a kind of tiling puzzle; the input is a set of cards, and a part of a pipe is drawn on each card. For a given set of cards, we arrange them and connect the pipes. We have to connect all pipes without creating any local loop. While ordinary tiling puzzles, like jigsaw puzzles, ask to arrange the tiles with local consistency, pipe puzzles ask to join all pipes. We first show that the pipe puzzle is NP-complete in general even if the goal shape is quite restricted. We also investigate restricted cases and show some polynomial-time algorithms.

Secure sets and defensive alliances in graphs are studied. They are sets of vertices that are safe in some senses. In this paper, we first present a fixed-parameter algorithm for finding a small secure set, whose running time is much faster than the previously known one. We then present improved bound on the smallest sizes of defensive alliances and secure sets for hypercubes. These results settle some open problems paused recently.

In this letter, we present a new lower bound for the treewidth of a graph in terms of the second smallest eigenvalue of its Laplacian matrix. Our bound slightly improves the lower bound given by Chandran and Subramanian [Inf. Process. Lett., 87 (2003)].

In this paper, we study BALL COLLECTING WITH LIMITED ENERGY, which is a problem of scheduling robots with limited energy confined to a line to catch moving balls that eventually cross the line. For this problem, we show the NP-completeness of the general case and some algorithmic results for some cases with a small number of robots.

In this note, we consider the problem of finding a step-by-step transformation between two longest increasing subsequences in a sequence, namely LONGEST INCREASING SUBSEQUENCE RECONFIGURATION. We give a polynomial-time algorithm for deciding whether there is a reconfiguration sequence between two longest increasing subsequences in a sequence. This implies that INDEPENDENT SET RECONFIGURATION and TOKEN SLIDING are polynomial-time solvable on permutation graphs, provided that the input two independent sets are largest among all independent sets in the input graph. We also consider a special case, where the underlying permutation graph of an input sequence is bipartite. In this case, we give a polynomial-time algorithm for finding a shortest reconfiguration sequence (if it exists).

This paper presents an efficient algorithm to generate all (unordered) rooted trees with exactly n vertices including exactly k leaves. There are known results on efficient enumerations of some classes of graphs embedded on a plane, for instance, biconnected and triconnected triangulations [3],[6], and floorplans [4]. On the other hand, it is difficult to enumerate a class of graphs without a fixed embedding. The paper is on enumeration of rooted trees without a fixed embedding. We already proposed an algorithm to generate all “ordered” trees with n vertices including k leaves [11], while the algorithm cannot seem to efficiently generate all (unordered) rooted trees with n vertices including k leaves. We design a simple tree structure among such trees, then by traversing the tree structure we generate all such trees in constant time per tree in the worst case. By repeatedly applying the algorithm for each k=1,2, ...,n-1, we can also generate all rooted trees with exactly n vertices.