A chordal graph is a graph which contains no chordless cycle of at least four edges as an induced subgraph. The class of chordal graphs contains many famous graph classes such as trees, interval graphs, and split graphs, and is also a subclass of perfect graphs. In this paper, we address the problem of enumerating all labeled chordal graphs included in a given graph. We think of some variations of this problem. First we introduce an algorithm to enumerate all connected labeled chordal graphs in a complete graph of n vertices. Next, we extend the algorithm to an algorithm to enumerate all labeled chordal graphs in a n-vertices complete graph. Then, we show that we can use, with small changes, these algorithms to generate all (connected or not necessarily connected) labeled chordal graphs in arbitrary graph. All our algorithms are based on reverse search method, and time complexities to generate a chordal graph are O(1), and also O(1) delay. Additionally, we present an algorithm to generate every clique of a given chordal graph in constant time. Using these algorithms we obtain combinatorial Gray code like sequences for these graph structures in which the differences between two consecutive graphs are bounded by a constant size.

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).

We investigate connected proper interval graphs without vertex labels. We first give the number of connected proper interval graphs of n vertices. Using this result, a simple algorithm that generates a connected proper interval graph uniformly at random up to isomorphism is presented. Finally an enumeration algorithm of connected proper interval graphs is proposed. The algorithm is based on reverse search, and it outputs each connected proper interval graph in (O)1 time.

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 Voronoi game is a two-person perfect information game modeling a competitive facility location. The original version of the game is played on a continuous domain. Only two special cases (1-dimensional case and 1-round case) have been extensively investigated. Recently, the discrete Voronoi game of which the game arena is given as a graph was introduced. In this note, we give a complete analysis of the discrete Voronoi game on a path. There are drawing strategies for both the first and the second players, except for some trivial cases.