Recognition Algorithm for Probe Interval 2-Trees

Recognition of probe interval graphs has been studied extensively. Recognition algorithms of probe interval graphs can be broken down into two types of problems: partitioned and non-partitioned. A partitioned recognition algorithm includes the probe and nonprobe partition of the vertices as part of the input, where a non-partitioned algorithm does not include the partition. Partitioned probe interval graphs can be recognized in linear-time in the edges, whereas non-partitioned probe interval graphs can be recognized in polynomial-time. Here we present a non-partitioned recognition algorithm for 2-trees, an extension of trees, that are probe interval graphs. We show that this algorithm runs in O(m) time, where m is the number of edges of a 2-tree. Currently there is no algorithm that performs as well for this problem

Recognition Algorithms for 2-Tree Probe Interval Graphs

This thesis focuses on looking at a particular set of graphs and recognizing if a given graph has certain properties that would make it belong in this family, here called 2-tree Probe Interval Graphs. For these graphs, we create an algorithm to run on a coded script that recursively runs criteria through an input graph from its matrix representation to check the 2-path, and will output either a success that our graph is a 2-tree Probe Interval Graph, or failure if it is not. After the creation of this algorithm, a complexity analysis for the algorithm will be developed, as well as the implementation of di_erent search criteria to hopefully reduce the complexity by some polynomial factor. The recognition for our set of graphs follows to the conceptual idea that triangles are built upon each other in a fashion of adding one vertex and two edges to a previous triangle in the graph. Each new triangle is added to an existing triangle and recursively builds the graph where the new vertex neighbors strictly two vertices with an existing triangle, creating a recursively de_ned 2-path