### The Parameterized Complexity of k-Edge Induced Subgraphs [chapter]

Bingkai Lin, Yijia Chen
2012 Lecture Notes in Computer Science
We prove that finding a k-edge induced subgraph is fixedparameter tractable, thereby answering an open problem of Leizhen Cai [2] . Our algorithm is based on several combinatorial observations, Gauss' famous Eureka theorem [1] , and a generalization of the wellknown fpt-algorithm for the model-checking problem for first-order logic on graphs with locally bounded tree-width due to Frick and Grohe [13] . On the other hand, we show that two natural counting versions of the problem are hard. Hence,
more » ... the k-edge induced subgraph problem is one of the very few known examples in parameterized complexity that are easy for decision while hard for counting. p-Edge-Induced-Subgraph Instance: A graph G and k ∈ N. Parameter: k. Problem: Decide whether G contains a k-edge induced subgraph. As the main result of our paper, we show that p-Edge-Induced-Subgraph is fixed-parameter tractable. In fact, there are special cases of p-Edge-Induced-Subgraph whose fixed-parameter tractability has been known for a while. Since ⋆ Full version available at http://arxiv.org/abs/1105.0477 we can define a k-edge induced subgraph by a first-order sentence, using logic machinery, it can be shown that p-Edge-Induced-Subgraph is fixed-parameter tractable if the graph G has bounded tree-width [8], bounded local tree-width [13], etc., or most generally locally bounded expansion [10] . Unfortunately, the class of all graphs containing a k-edge induced subgraph does not possess any of these bounded measures. As another previously known case, using his Random Separation method [5] and Ramsey's Theorem, Cai [4] gave a very nice combinatorial algorithm that solves p-Edge-Induced-Subgraph when the parameter k is a triangular number, i.e., k = m 2 for some m ∈ N. However, it looks very difficult to adapt Cai's algorithm to handle arbitrary k. Therefore neither logic nor combinatorial approach so far seems to be sufficient to settle the complexity of p-Edge-Induced-Subgraph by its own. So our fpt-algorithm is a rather tricky combination of these two methods. Our approach As just mentioned, our starting pointing is that the existence of a k-edge induced subgraph can be characterized by a sentence of first-order logic (FO) which depends on k only. It is a well-known result of Frick and Grohe [13] that the model-checking problem for FO on graphs of bounded local tree-width is fixedparameter tractable. The local tree-width for a graph is a function bounding the tree-width of the induced subgraphs on the neighborhoods within a certain radius of every vertex. For instance, bounded-degree graphs have bounded local tree-width. These give immediately the fixed-parameter tractability of p-Edge-Induced-Subgraph on graphs with bounded degree 1 . With some more efforts, the above result can be extended to graphs G with degree bounded by a function of the parameter k. In that case, we can say the degree deg(v) of each vertex v is sufficiently small. The corresponding fptalgorithm generalizes Frick and Grohe's Theorem to graphs with local tree-width bounded by a function of both the radius of the neighborhoods and an additional parameter. As a dual, if deg(v) of each vertex v in G is sufficiently large, or more precisely, the complement of G has degree bounded by a function of k, then we can decide p-Edge-Induced-Subgraph in fpt time, too. Moving one step further, we consider graphs in which each deg(v) is either sufficiently small or sufficiently large, e.g., an n-star. We call such graphs degreeextreme. Using the same logic machinery as above, we are able to show the fixed-parameter tractability of p-Edge-Induced-Subgraph on degree-extreme graphs. Assume that the graph G is not degree-extreme, i.e., there exists a vertex v 0 whose degree is neither sufficiently small nor sufficiently large. We partition the vertex set of G into two sets V 1 and V 2 , where V 1 contains all vertices adjacent to v 0 and V 2 the remaining vertices. Then both V 1 and V 2 are relatively large. Note