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Forbidden Minor Characterizations For 4-Searchable Graphs

Final Report Summary - FORMI 4-SEG (Forbidden Minor Characterizations For 4-Searchable Graphs)

Edge searching is a pursuit evasion game played on graphs. The aim is to capture an intruder in a system of caves which can be represented as a graph where junctions correspond to vertices and tunnels correspond to edges. For this end we launch a group of searchers into the system. The intruder is invisible and he can move along the edges with an unbounded speed; as long as there are no searchers located on his trajectory. Furthermore, the intruder has complete information about the locations of the searchers. The goal here is to construct a search plan and to find the minimum size of the search team that will guarantee the capture of the intruder.

Let G=(V, E) denote a graph where V is its vertex set and E is its edge set. We assume that every edge of G is contaminated initially and our aim is to clean the whole graph by a sequence of steps. First we place a given set of k searchers/cleaners on a subset of V. We can place as many searchers as we like on each vertex. Each step of an edge search strategy consists of one of the following moves: removing a searcher from one vertex and placing it on another vertex, or sliding a searcher from a vertex u along an edge e=uv to an adjacent vertex v.

An edge search strategy is a combination of the moves so that the state of all edges being simultaneously clean is achieved, in which case we say that the graph is cleaned. The problem becomes cleaning the graph using the least number of searchers. This number is a graph invariant called the edge search number of the graph and is denoted s(G).

If a searcher slides along an edge e = uv from u to v, then e is cleaned if either (i) another searcher is stationed at u, or (ii) all other edges incident to u are already clean. If a searcher is stationed at a vertex v, then we say that v is guarded. If a path does not contain any guarded vertex, then it is called an unguarded path. If there is an unguarded path that contains one endpoint of a contaminated edge and one endpoint of a cleaned edge e, then e gets recontaminated.

The decision version of the problem is computationally hard: Given a graph G and an integer k, it has been shown, by Megiddo et. al, in 1988 that deciding whether G can be searched by k searchers is NP-complete. It can be solved in polynomial time when the problem is restricted to trees.

One of the major problems of edge search is to characterize the graphs G such that s(G) is at most k for a fixed positive integer k, in which case we say that G is k-searchable. Given a graph G consider the following ways of reducing it (1) delete an edge, (2) contract an edge, (3) delete an isolated vertex. Any graph H that can be produced from G by successive application of these reductions is called a minor of G. It is known that the edge search number is closed under the taking of minors. Thus, if G contains H as a minor, then s(H) is at most s(G).

We say that a graph H is a forbidden minor for a k-searchable graph, if k < s(H) and if any proper minor F of H has search number at most k. For fixed k, the set of forbidden minors for k-searchable graphs is called the obstruction set or the Kuratowski set.

The graph minor theory built by Robertson and Seymour implies that the obstruction set is finite for an invariant that is closed under the taking of minors. On the other hand no general method is known for constructing an obstruction set. In graph searching, the graphs that are 1,2 or 3-searchable have been completely characterized by Megiddo, Hakimi, Garey, Johnsohn and Papadimitriou in 1988. Characterizing k-searchable graphs where k is greater than 3 is left as an open problem since then.

The main goal in this project consists in the algorithmic construction of the forbidden minors for k-searchable graphs when k=4 and use this construction technique in order to generalize the result to any k greater than 4. The most innovative nature of our approach to the problem is that we target to find a recursive constructive method for the characterization of these forbidden minors.

On 4-searchable graphs the only existing result is for outerplanar graphs (Yasar, 2008). In this project we start with constructing the forbidden minors for series-parallel graphs for which outerplanar graphs form a maximal subclass.

First, we have completed our results on the construction of the forbidden minors of series-parallel graphs. Thus we have found the complete list of forbidden minors for biconnected 4-searchable series-parallel graphs.

Next, we reformulated our construction so that it gives us an algorithmic construction rather than a case specific one. Our technique can be used to construct algorithmically the forbidden minors for k-searchable series-parallel graphs, for each k less than 5, by giving first the families of graphs that will be used as building blocks, and then by taking the proper combinations.

Furthermore, by using two different ways of extending the forbidden minors for biconnected (k-2)-searchable series parallel graphs we construct the forbidden minors for biconnected k-searchable series parallel graphs.

Forbidden minor constructions fall into the problems under topological graph theory. Here we propose a general operation called k-sums, which is closely related to the connected-sum operation. The k-sums method is a generalization of the k-clique sum method which has been proven to be useful in the construction of the characterization of minor closed families. In this project, we show how k-sums can be used for the construction of the forbidden minors for k-searchable graphs.

The main innovative nature of our results is that they give an explicit and algorithmic construction of forbidden minors for 4-searchable graphs for some graph classes. We further introduce several algorithmic methods to extend these constructions to k larger than 4.

The results of this project are important mainly in achieving a method that is recursive in nature and independent of the specific case under consideration. The methodology we construct can be used for the characterization of forbidden minors for any minor closed family.

For future research, the obstruction sets obtained in this project can be used to give the exact explicit construction of 4-searchable graphs. The main potential impact of this result will be the generalization of the construction of the forbidden minors for 4-searchable graphs to the construction of the forbidden minors for k-searchable graphs where k is larger than 4.

This is a joint work of Dr. Danny Dyer (Memorial University of Newfoundland, Canada) and Dr. Dariusz Dereniowski (Gdansk University of Technology, Poland).