102 research outputs found
Proximity Search for Maximal Subgraph Enumeration
International audienc
Efficient enumeration of maximal split subgraphs and sub-cographs and related classes
In this paper, we are interested in algorithms that take in input an
arbitrary graph , and that enumerate in output all the (inclusion-wise)
maximal "subgraphs" of which fulfil a given property . All over this
paper, we study several different properties , and the notion of subgraph
under consideration (induced or not) will vary from a result to another.
More precisely, we present efficient algorithms to list all maximal split
subgraphs, sub-cographs and some subclasses of cographs of a given input graph.
All the algorithms presented here run in polynomial delay, and moreover for
split graphs it only requires polynomial space. In order to develop an
algorithm for maximal split (edge-)subgraphs, we establish a bijection between
the maximal split subgraphs and the maximal independent sets of an auxiliary
graph. For cographs and some subclasses , the algorithms rely on a framework
recently introduced by Conte & Uno called Proximity Search. Finally we consider
the extension problem, which consists in deciding if there exists a maximal
induced subgraph satisfying a property that contains a set of prescribed
vertices and that avoids another set of vertices. We show that this problem is
NP-complete for every "interesting" hereditary property . We extend the
hardness result to some specific edge version of the extension problem
All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs
We describe algorithms, based on Avis and Fukuda's reverse search paradigm,
for listing all maximal independent sets in a sparse graph in polynomial time
and delay per output. For bounded degree graphs, our algorithms take constant
time per set generated; for minor-closed graph families, the time is O(n) per
set, and for more general sparse graph families we achieve subquadratic time
per set. We also describe new data structures for maintaining a dynamic vertex
set S in a sparse or minor-closed graph family, and querying the number of
vertices not dominated by S; for minor-closed graph families the time per
update is constant, while it is sublinear for any sparse graph family. We can
also maintain a dynamic vertex set in an arbitrary m-edge graph and test the
independence of the maintained set in time O(sqrt m) per update. We use the
domination data structures as part of our enumeration algorithms.Comment: 10 page
Efficient Enumeration of Dominating Sets for Sparse Graphs
A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in O(k) time per solution using O(n + m) space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine
Neighborhood Inclusions for Minimal Dominating Sets Enumeration: Linear and Polynomial Delay Algorithms in P_7 - Free and P_8 - Free Chordal Graphs
In [M. M. Kant\'e, V. Limouzy, A. Mary, and L. Nourine. On the enumeration of
minimal dominating sets and related notions. SIAM Journal on Discrete
Mathematics, 28(4):1916-1929, 2014] the authors give an delay
algorithm based on neighborhood inclusions for the enumeration of minimal
dominating sets in split and -free chordal graphs. In this paper, we
investigate generalizations of this technique to -free chordal graphs for
larger integers . In particular, we give and delays
algorithms in the classes of -free and -free chordal graphs. As for
-free chordal graphs for , we give evidence that such a technique
is inefficient as a key step of the algorithm, namely the irredundant extension
problem, becomes NP-complete.Comment: 16 pages, 3 figure
FPT algorithms for finding near-cliques in c-closed graphs
Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of c-closed graphs and proved that maximal clique enumeration is fixed-parameter tractable with respect to c. In practice, due to noise in data, one wishes to actually discover “near-cliques”, which can be characterized as cliques with a sparse subgraph removed. In this work, we prove that many different kinds of maximal near-cliques can be enumerated in polynomial time (and FPT in c) for c-closed graphs. We study various established notions of such substructures, including k-plexes, complements of bounded-degeneracy and bounded-treewidth graphs. Interestingly, our algorithms follow relatively simple backtracking procedures, analogous to what is done in practice. Our results underscore the significance of the c-closed graph class for theoretical understanding of social network analysis
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