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LNCS, volume 10236; International audience; An independent 2-clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside. In this paper, we study the complexity of finding an independent 2-clique of maximum size in several graph classes and we compare its complexity with the complexity of maximum independent set. We prove that this problem is NP-hard on apex graphs, APX-hard on line graphs, not n1/2−ϵ-approximable on bipartite graphs and not n1−ϵ-approximable on split graphs, while it is polynomial-time solvable on graphs of bounded degree and their complements, graphs of bounded treewidth, planar graphs, (C3,C6)-free graphs, threshold graphs, interval graphs and cographs.

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size $\mathcal {O}(k^{3/2})$ in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios $\mathcal {O}(\log{k})$ on planar graphs and $\mathcal {O}(\sqrt{k}\log{k})$ on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.

We investigate the structural and algorithmic properties of 2-community structure in graphs introduced by Olsen [A general view on computing communities, Mathematical Social Sciences 66(3):331--336, 2013]. A 2-community structure is a partition of vertex set into two parts such that for each vertex of the graph number of neighbours in/outside own part is in correlation with sizes of parts. We show that every 3-regular graph has a 2-community structure which can be found in polynomial time, even if the subgraphs induced by each partition must be connected. We introduce a concept of a 2-weak community and prove that it is NP-complete to find a balanced 2-weak community structure in general graphs evenwith additional request of connectivity for both parts. On the other hand, the problem can be solved in polynomial time in graphs of degree at most 3.

Given an undirected graph on n vertices with weights on its edges, Min WCF (p) consists of computing a covering forest of minimum weight such that each of its tree components contains at least p vertices. It has been proved that Min WCF (p) is NP-hard for any p≥4 (Imielinska et al. (1993) [10]) but (2−1n)-approximable (Goemans and Williamson (1995) [9]). While Min WCF(2) is polynomial-time solvable, already the unweighted version of Min WCF(3) is NP-hard even on planar bipartite graphs of maximum degree 3. We prove here that for any p≥4, the unweighted version is NP-hard, even for planar bipartite graphs of maximum degree 3; moreover, the unweighted version for any p≥3 has no ptas for bipartite graphs of maximum degree 3. The latter theorem is the first-ever APX-hardness result on this problem. On the other hand, we show that Min WCF (p) is polynomial-time solvable on graphs with bounded treewidth, and for any p bounded by O(lognloglogn) it has a ptas on planar graphs.

We consider the degree-preserving spanning tree (DPST) problem: Given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph-theoretical translation of a problem arising in the system-theoretical context of identifiability in networks, a concept which has applications in, for example, water distribution networks and electrical networks. We show that the DPST problem is NP-complete, even when restricted to split graphs or bipartite planar graphs, but that it can be solved in polynomial time for graphs with a bounded asteroidal number and for graphs with a bounded treewidth. For the class of interval graphs, we give a linear time algorithm. For the class of cocomparability graphs, we give an O(n4) algorithm. Furthermore, we present linear time approximation algorithms for planar graphs of a worst-case performance ratio of 1 - e for every e > 0.

A team of mobile agents, called guards, tries to keep an intruder out of an assigned area by blocking all possible attacks. In a graph model for this setting, the guards and the intruder are located on the vertices of a graph, and they move from node to node via connecting edges. The area protected by the guards is an induced subgraph of the given graph. We investigate the algorithmic aspects of the guarding problem, which is to find the minimum number of guards sufficient to patrol the area. We show that the guarding problem is PSPACE-hard and provide a set of approximation algorithms. All approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win. This variant of the game appears to be a 2-approximation for the guarding problem, and for graphs without cycles of length 5 the minimum number of required guards in both games coincides. We give a polynomial time algorithm for solving the one-step guarding problem in graphs of bounded treewidth, and complement this result by showing that the problem is W[1]-hard parameterized by the treewidth of the input graph. We also show that the problem is fixed parameter tractable (FPT) parameterized by the treewidth and maximum degree of the input graph. Finally, we turn our attention to a large class of sparse graphs, including planar graphs and graphs of bounded genus, namely apex-minor-free graphs. We prove that the one-step guarding problem is FPT and possess a PTAS on apex-minor-free graphs.

A t-spanner of a graph G is its spanning subgraph S such that the distance between every pair of vertices in S is at most t times their distance in G. The sparsestt-spanner problem asks to find, for a given graph G and an integer t, a t-spanner of G with the minimum number of edges. The problem is known to be NP-hard for all t≥2, and, even more, it is NP-hard to approximate it with ratio O(logn) for every t≥2. For t≥5, the problem remains NP-hard for planar graphs and the approximability status of the problem on planar graphs was open. We resolve this open issue by showing that the sparsestt-spanner problem admits the efficient polynomial time approximation scheme (EPTAS) for every t≥1. Our result holds for a much wider class of graphs, namely, the class of apex-minor-free graphs, which contains the classes of planar and bounded genus graphs. Moreover, it is possible to extend our results to weighted apex-minor free graphs, when the maximum edge weight is bounded by some constant.

Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs. Therefore, unless P=NP, these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.

International audience; We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331–336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V|−3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.