12,468 research outputs found
Combinatorial Generalizations of Sieve Methods and Characterizing Hamiltonicity via Induced Subgraphs
A sieve method is in effect an application of the inclusion-exclusion counting principle, and the estimation methods to avoid computing the explicit formula. Sieve methods have been used in number theory for over a hundred years. These methods have been modified to make use of the structure of integer-like objects; producing better estimates and providing more use cases. The first part of the thesis aims to analyze and use the analogues of number theoretic sieves in combinatorial contexts. This part consists of my work with Yu-Ru Liu in Chapters 2 and 3. We focus on two sieve methods: the Turán sieve (introduced by Liu and Murty in 2005) and the Selberg sieve (independently generalized by Wilson in 1969 and Chow in 1998 with slightly different formulations). Some comparisons and applications of these sieves are discussed. In particular, we apply the combinatorial Turán sieve to count labelled graphs and we apply the combinatorial Selberg sieve to count subspaces of finite
spaces.
Finding sufficient conditions for Hamiltonicity in graphs is a classical topic, where the difficulty is bracketed by the NP-hardness of the associated decision problem. The second part of the thesis, consisting of Chapter 4, aims to characterize Hamiltonicity by means of induced subgraphs. The results in this chapter are based on the paper "Minimal induced subgraphs of two classes of 2-connected non-Hamiltonian graphs." Discrete Mathematics, 345(7):112869, 2022, co-authored with Joseph Cheriyan, Sepehr Hajebi, and Sophie Spirkl. We study induced subgraphs and conditions for Hamiltonicity. In particular, we characterize
the minimal 2-connected non-Hamiltonian split graphs and the minimal 2-connected non-Hamiltonian triangle-free graphs
Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs
for the class of interval graphs. We give a linear-time algorithm to find one
in any graph that is not an interval graph. Tucker characterized the minimal
forbidden submatrices of binary matrices that do not have the consecutive-ones
property. We give a linear-time algorithm to find one in any binary matrix that
does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th
International Workshop on Graph-Theoretic Concepts in Computer Scienc
Communities as Well Separated Subgraphs With Cohesive Cores: Identification of Core-Periphery Structures in Link Communities
Communities in networks are commonly considered as highly cohesive subgraphs
which are well separated from the rest of the network. However, cohesion and
separation often cannot be maximized at the same time, which is why a
compromise is sought by some methods. When a compromise is not suitable for the
problem to be solved it might be advantageous to separate the two criteria. In
this paper, we explore such an approach by defining communities as well
separated subgraphs which can have one or more cohesive cores surrounded by
peripheries. We apply this idea to link communities and present an algorithm
for constructing hierarchical core-periphery structures in link communities and
first test results.Comment: 12 pages, 2 figures, submitted version of a paper accepted for the
7th International Conference on Complex Networks and Their Applications,
December 11-13, 2018, Cambridge, UK; revised version at
http://141.20.126.227/~qm/papers
Finding Induced Subgraphs via Minimal Triangulations
Potential maximal cliques and minimal separators are combinatorial objects
which were introduced and studied in the realm of minimal triangulations
problems including Minimum Fill-in and Treewidth. We discover unexpected
applications of these notions to the field of moderate exponential algorithms.
In particular, we show that given an n-vertex graph G together with its set of
potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G|
* n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a
given graph F of treewidth t, to decide if G contains an induced subgraph
isomorphic to F. Combined with an improved algorithm enumerating all potential
maximal cliques in time O(1.734601^n), this yields that both problems are
solvable in time 1.734601^n * n^(O(t)).Comment: 14 page
On maximal chain subgraphs and covers of bipartite graphs
In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem.
The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time
Subgraph covers -- An information theoretic approach to motif analysis in networks
Many real world networks contain a statistically surprising number of certain
subgraphs, called network motifs. In the prevalent approach to motif analysis,
network motifs are detected by comparing subgraph frequencies in the original
network with a statistical null model. In this paper we propose an alternative
approach to motif analysis where network motifs are defined to be connectivity
patterns that occur in a subgraph cover that represents the network using
minimal total information. A subgraph cover is defined to be a set of subgraphs
such that every edge of the graph is contained in at least one of the subgraphs
in the cover. Some recently introduced random graph models that can incorporate
significant densities of motifs have natural formulations in terms of subgraph
covers and the presented approach can be used to match networks with such
models. To prove the practical value of our approach we also present a
heuristic for the resulting NP-hard optimization problem and give results for
several real world networks.Comment: 10 pages, 7 tables, 1 Figur
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes
Many fixed-parameter tractable algorithms using a bounded search tree have
been repeatedly improved, often by describing a larger number of branching
rules involving an increasingly complex case analysis. We introduce a novel and
general search strategy that branches on the forbidden subgraphs of a graph
class relaxation. By using the class of -sparse graphs as the relaxed
graph class, we obtain efficient bounded search tree algorithms for several
parameterized deletion problems. We give the first non-trivial bounded search
tree algorithms for the cograph edge-deletion problem and the trivially perfect
edge-deletion problems. For the cograph vertex deletion problem, a refined
analysis of the runtime of our simple bounded search algorithm gives a faster
exponential factor than those algorithms designed with the help of complicated
case distinctions and non-trivial running time analysis [21] and computer-aided
branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and
Applications (DMAA
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