2,283 research outputs found
Parameterized Aspects of Strong Subgraph Closure
Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the Strong F-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In Strong F-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to destroy that strong copy of F in G.
We study Strong F-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when F =P_3. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization k - mu(G), where mu(G) is the maximum matching size of G. We conclude with some results on the parameterization of Strong F-closure by the number of edges of G that are not selected as strong
Spectrally degenerate graphs: Hereditary case
It is well known that the spectral radius of a tree whose maximum degree is D
cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar
graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally,
for all d-degenerate graphs, where the corresponding upper bound is sqrt{4dD}.
Following this, we say that a graph G is spectrally d-degenerate if every
subgraph H of G has spectral radius at most sqrt{d.Delta(H)}. In this paper we
derive a rough converse of the above-mentioned results by proving that each
spectrally d-degenerate graph G contains a vertex whose degree is at most
4dlog_2(D/d) (if D>=2d). It is shown that the dependence on D in this upper
bound cannot be eliminated, as long as the dependence on d is subexponential.
It is also proved that the problem of deciding if a graph is spectrally
d-degenerate is co-NP-complete.Comment: Updated after reviewer comments. 14 pages, no figure
Spectral radius of finite and infinite planar graphs and of graphs of bounded genus
It is well known that the spectral radius of a tree whose maximum degree is
cannot exceed . In this paper we derive similar bounds for
arbitrary planar graphs and for graphs of bounded genus. It is proved that a
the spectral radius of a planar graph of maximum vertex degree
satisfies . This result is
best possible up to the additive constant--we construct an (infinite) planar
graph of maximum degree , whose spectral radius is . This
generalizes and improves several previous results and solves an open problem
proposed by Tom Hayes. Similar bounds are derived for graphs of bounded genus.
For every , these bounds can be improved by excluding as a
subgraph. In particular, the upper bound is strengthened for 5-connected
graphs. All our results hold for finite as well as for infinite graphs.
At the end we enhance the graph decomposition method introduced in the first
part of the paper and apply it to tessellations of the hyperbolic plane. We
derive bounds on the spectral radius that are close to the true value, and even
in the simplest case of regular tessellations of type we derive an
essential improvement over known results, obtaining exact estimates in the
first order term and non-trivial estimates for the second order asymptotics
Near-colorings: non-colorable graphs and NP-completeness
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned
into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of
V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus
on complexity aspects of such colorings when l=2,3. More precisely, we prove
that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3,
either every planar graph with girth at least g is (k,j)-colorable or it is
NP-complete to determine whether a planar graph with girth at least g is
(k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine
whether a planar graph that is either (0,0,0)-colorable or
non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit
non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar
graphs with girth 7
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration
We continue research into a well-studied family of problems that ask whether
the vertices of a graph can be partitioned into sets and~, where is
an independent set and induces a graph from some specified graph class
. We let be the class of -degenerate graphs. This
problem is known to be polynomial-time solvable if (bipartite graphs) and
NP-complete if (near-bipartite graphs) even for graphs of maximum degree
. Yang and Yuan [DM, 2006] showed that the case is polynomial-time
solvable for graphs of maximum degree . This also follows from a result of
Catlin and Lai [DM, 1995]. We consider graphs of maximum degree on
vertices. We show how to find and in time for , and in
time for . Together, these results provide an algorithmic
version of a result of Catlin [JCTB, 1979] and also provide an algorithmic
version of a generalization of Brook's Theorem, which was proven in a more
general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007].
Moreover, the two results enable us to complete the complexity classification
of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex
colouring reconfiguration graph between two given -colourings of a graph
of maximum degree
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