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).

A hinge vertex is a vertex in an undirected graph such that there exist two vertices whose removal makes the distance between them longer than before. Identifying hinge vertices in a graph can help detect critical nodes in communication network systems, which is useful for making them more stable. For finding them, an O(n3) time algorithm was developed for a simple graph, and, linear time algorithms were developed for interval and permutation graphs, respectively. Recently, the maximum detour hinge vertex problem is defined by Honma et al. For a hinge vertex u in a graph, the detour degree of u is the largest value of distance between any pair of x and y (x and y are adjacent to u) by removing u. A hinge vertex with the largest detour degree in G is defined as the maximum detour hinge vertex of G. This problem is motivated by practical applications, such as network stabilization with a limited cost, i.e., by enhancing the reliability of the maximum detour hinge vertex, the stability of the network is much improved. We previously developed an O(n2) time algorithm for solving this problem on an interval graph. In this study, we propose an algorithm that identifies the maximum detour hinge vertex on a permutation graph in O(n2) time, where n is the number of vertices in the graph.

Ordered Binary Decision Diagrams (OBDDs for short) are popular dynamic data structures for Boolean functions. In some modern applications, we have to handle such huge graphs that the usual explicit representations by adjacency lists or adjacency matrices are infeasible. To deal with such huge graphs, OBDD-based graph representations and algorithms have been investigated. Although the size of OBDD representations may be large in general, it is known to be small for some special classes of graphs. In this paper, we show upper bounds and lower bounds of the size of OBDDs representing some intersection graphs such as bipartite permutation graphs, biconvex graphs, convex graphs, (2-directional) orthogonal ray graphs, and permutation graphs.

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2log 6n) time for n-vertex graphs with maximum degree at most three.

Let Gs=(Vs, Es) be a simple connected graph. A vertex v ∈ Vs is an articulation vertex if deletion of v and its incident edges from Gs disconnects the graph into at least two connected components. Finding all articulation vertices of a given graph is called the articulation vertex problem. A vertex u ∈ Vs is called a hinge vertex if there exist any two vertices x and y in Gs whose distance increase when u is removed. Finding all hinge vertices of a given graph is called the hinge vertex problem. These problems can be applied to improve the stability and robustness of communication network systems. In this paper, we propose linear time algorithms for the articulation vertex problem and the hinge vertex problem of circular permutation graphs.

PREIMAGE CONSTRUCTION problem by Kratsch and Hemaspaandra naturally arose from the famous graph reconstruction conjecture. It deals with the algorithmic aspects of the conjecture. We present an O(n8) time algorithm for PREIMAGE CONSTRUCTION on permutation graphs and an O(n4(n+m)) time algorithm for PREIMAGE CONSTRUCTION on distance-hereditary graphs, where n is the number of graphs in the input, and m is the number of edges in a preimage. Since each graph of the input has n-1 vertices and O(n2) edges, the input size is O(n3) (, or O(nm)). There are polynomial time isomorphism algorithms for permutation graphs and distance-hereditary graphs. However the number of permutation (distance-hereditary) graphs obtained by adding a vertex to a permutation (distance-hereditary) graph is generally exponentially large. Thus exhaustive checking of these graphs does not achieve any polynomial time algorithm. Therefore reducing the number of preimage candidates is the key point.

The spanning tree problem is to find a tree that connects all the vertices of G. This problem has many applications, such as electric power systems, computer network design and circuit analysis. Klein and Stein demonstrated that a spanning tree can be found in O(log n) time with O(n+m) processors on the CRCW PRAM. In general, it is known that more efficient parallel algorithms can be developed by restricting classes of graphs. Circular permutation graphs properly contain the set of permutation graphs as a subclass and are first introduced by Rotem and Urrutia. They provided O(n2.376) time recognition algorithm. Circular permutation graphs and their models find several applications in VLSI layout. In this paper, we propose an optimal parallel algorithm for constructing a spanning tree on circular permutation graphs. It runs in O(log n) time with O(n/log n) processors on the EREW PRAM.

A tree 3-spanner T of a graph G is a spanning tree of G such that the distance between any two vertices in T is at most 3 times of their distance in G. Madanlal et al. have presented an O(n + m) time algorithm for finding a tree 3-spanner of a permutation graph. However, the complexity of their algorithm is not optimal, and their algorithm can not be easily parallelized. In this paper, we will propose an improved algorithm to solve the same problem in O(n) time. Moreover, our algorithm can be easily parallelized so that a tree 3-spanner of a permutation graph can be found in O(log n) time with processors on the EREW PRAM computational model.