740 research outputs found
Parameterised and fine-grained subgraph counting, modulo 2
Given a class of graphs H, the problem ⊕Sub(H) is defined as follows. The input is a graph H ∈ H together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes H the problem ⊕Sub(H) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|) · |G| O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H) is FPT if and only if the class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ H can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes H, and (II) all tree pattern classes, i.e., all classes H such that every H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I)
Structure and colour in triangle-free graphs
Motivated by a recent conjecture of the first author, we prove that every
properly coloured triangle-free graph of chromatic number contains a
rainbow independent set of size . This is sharp up to
a factor . This result and its short proof have implications for the related
notion of chromatic discrepancy.
Drawing inspiration from both structural and extremal graph theory, we
conjecture that every triangle-free graph of chromatic number contains
an induced cycle of length as . Even if
one only demands an induced path of length , the
conclusion would be sharp up to a constant multiple. We prove it for regular
girth graphs and for girth graphs.
As a common strengthening of the induced paths form of this conjecture and of
Johansson's theorem (1996), we posit the existence of some such that for
every forest on vertices, every triangle-free and induced -free
graph has chromatic number at most . We prove this assertion with
`triangle-free' replaced by `regular girth '.Comment: 12 pages; in v2 one section was removed due to a subtle erro
Vertex-Coloring with Star-Defects
Defective coloring is a variant of traditional vertex-coloring, according to
which adjacent vertices are allowed to have the same color, as long as the
monochromatic components induced by the corresponding edges have a certain
structure. Due to its important applications, as for example in the
bipartisation of graphs, this type of coloring has been extensively studied,
mainly with respect to the size, degree, and acyclicity of the monochromatic
components.
In this paper we focus on defective colorings in which the monochromatic
components are acyclic and have small diameter, namely, they form stars. For
outerplanar graphs, we give a linear-time algorithm to decide if such a
defective coloring exists with two colors and, in the positive case, to
construct one. Also, we prove that an outerpath (i.e., an outerplanar graph
whose weak-dual is a path) always admits such a two-coloring. Finally, we
present NP-completeness results for non-planar and planar graphs of bounded
degree for the cases of two and three colors
Colorings of complements of line graphs
Our purpose is to show that complements of line graphs enjoy nice coloring
properties. We show that for all graphs in this class the local and usual
chromatic numbers are equal. We also prove a sufficient condition for the
chromatic number to be equal to a natural upper bound. A consequence of this
latter condition is a complete characterization of all induced subgraphs of the
Kneser graph that have a chromatic number equal to its
chromatic number, namely . In addition to the upper bound, a lower bound
is provided by Dol'nikov's theorem, a classical result of the topological
method in graph theory. We prove the -hardness of deciding
the equality between the chromatic number and any of these bounds.
The topological method is especially suitable for the study of coloring
properties of complements of line graphs of hypergraphs. Nevertheless, all
proofs in this paper are elementary and we also provide a short discussion on
the ability for the topological methods to cover some of our results
Approximately counting and sampling small witnesses using a colourful decision oracle
In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the fine-grained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691)
Parameterised and Fine-Grained Subgraph Counting, Modulo 2
Given a class of graphs ?, the problem ?Sub(?) is defined as follows. The input is a graph H ? ? together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes ? the problem ?Sub(?) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)?|G|^O(1).
Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ?Sub(?) is FPT if and only if the class of allowed patterns ? is matching splittable, which means that for some fixed B, every H ? ? can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices.
Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes ?, and (II) all tree pattern classes, i.e., all classes ? such that every H ? ? is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I)
Nearly optimal independence oracle algorithms for edge estimation in hypergraphs
We study a query model of computation in which an n-vertex k-hypergraph can
be accessed only via its independence oracle or via its colourful independence
oracle, and each oracle query may incur a cost depending on the size of the
query. In each of these models, we obtain oracle algorithms to approximately
count the hypergraph's edges, and we unconditionally prove that no oracle
algorithm for this problem can have significantly smaller worst-case oracle
cost than our algorithms
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