Algorithms for Excluding Induced Subgraphs in Pt-free Graphs

In this paper, we explore a technique for partitioning graphs into smaller subgraphs based on their structure. This technique is crucial in developing algorithms for solving graph problems, such as clustering or network optimization. By utilizing the contraction-uncontraction method, we can divide each color class of an edge-colored graph into smaller parts while preserving its treewidth.
Imagine a graph as a complex network of roads connecting different cities. In this scenario, each city represents a vertex in the graph, and each road between them represents an edge. By partitioning these edges into smaller groups based on their connections, we can better understand how the network is structured.
The technique involves contracting connected components of one color class to a single vertex while maintaining the treewidth of the original graph. This process divides the maximum degree of each vertex by at most 3 and increases the treewidth of the resulting graph by at least a factor of tw(G)/(∆(G)qO(1)). After several iterations of this technique, we can extract an induced subgraph with at most three edges incident to every vertex that can recreate the original graph as a minor.
To illustrate this concept, imagine a road network with multiple intersections. By contracting each intersection to a single vertex, we can better understand how the roads are connected. However, this process also reduces the maximum degree of each vertex by at most 3. After several iterations of this technique, we can extract an induced subgraph with at most three edges incident to every vertex that can recreate the original road network as a minor.
In summary, this paper presents a technique for partitioning graphs into smaller subgraphs based on their structure, which is crucial in developing algorithms for solving graph problems. By utilizing the contraction-uncontraction method, we can better understand how graphs are structured and develop more efficient algorithms to solve graph problems.