146,585 research outputs found
Shortest-weight paths in random regular graphs
Consider a random regular graph with degree and of size . Assign to
each edge an i.i.d. exponential random variable with mean one. In this paper we
establish a precise asymptotic expression for the maximum number of edges on
the shortest-weight paths between a fixed vertex and all the other vertices, as
well as between any pair of vertices. Namely, for any fixed , we show
that the longest of these shortest-weight paths has about
edges where is the unique solution of the equation , for .Comment: 20 pages. arXiv admin note: text overlap with arXiv:1112.633
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
Parameterized Complexity of Simultaneous Planarity
Given input graphs , where each pair , with
shares the same graph , the problem Simultaneous Embedding With
Fixed Edges (SEFE) asks whether there exists a planar drawing for each input
graph such that all drawings coincide on . While SEFE is still open for the
case of two input graphs, the problem is NP-complete for [Schaefer,
JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We
show that SEFE is FPT with respect to plus the vertex cover number or the
feedback edge set number of the the union graph . Regarding the shared graph , we show that SEFE is NP-complete, even if
is a tree with maximum degree 4. Together with a known NP-hardness
reduction [Angelini et al., TCS 15], this allows us to conclude that several
parameters of , including the maximum degree, the maximum number of degree-1
neighbors, the vertex cover number, and the number of cutvertices are
intractable. We also settle the tractability of all pairs of these parameters.
We give FPT algorithms for the vertex cover number plus either of the first two
parameters and for the number of cutvertices plus the maximum degree, whereas
we prove all remaining combinations to be intractable.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
On the Parameterized Complexity of Sparsest Cut and Small-set Expansion Problems
We study the NP-hard \textsc{-Sparsest Cut} problem (SC) in which,
given an undirected graph and a parameter , the objective is to
partition vertex set into subsets whose maximum edge expansion is
minimized. Herein, the edge expansion of a subset is defined as
the sum of the weights of edges exiting divided by the number of vertices
in . Another problem that has been investigated is \textsc{-Small-Set
Expansion} problem (SSE), which aims to find a subset with minimum edge
expansion with a restriction on the size of the subset. We extend previous
studies on SC and SSE by inspecting their parameterized complexity. On
the positive side, we present two FPT algorithms for both SSE and 2SC
problems where in the first algorithm we consider the parameter treewidth of
the input graph and uses exponential space, and in the second we consider the
parameter vertex cover number of the input graph and uses polynomial space.
Moreover, we consider the unweighted version of the SC problem where is fixed and proposed two FPT algorithms with parameters treewidth and
vertex cover number of the input graph. We also propose a randomized FPT
algorithm for SSE when parameterized by and the maximum degree of the
input graph combined. Its derandomization is done efficiently.
\noindent On the negative side, first we prove that for every fixed integer
, the problem SC is NP-hard for graphs with vertex cover
number at most . We also show that SC is W[1]-hard when parameterized
by the treewidth of the input graph and the number~ of components combined
using a reduction from \textsc{Unary Bin Packing}. Furthermore, we prove that
SC remains NP-hard for graphs with maximum degree three and also graphs with
degeneracy two. Finally, we prove that the unweighted SSE is W[1]-hard for
the parameter
Complexity of planar signed graph homomorphisms to cycles
We study homomorphism problems of signed graphs. A signed graph is an
undirected graph where each edge is given a sign, positive or negative. An
important concept for signed graphs is the operation of switching at a vertex,
which is to change the sign of each incident edge. A homomorphism of a graph is
a vertex-mapping that preserves the adjacencies; in the case of signed graphs,
we also preserve the edge-signs. Special homomorphisms of signed graphs, called
s-homomorphisms, have been studied. In an s-homomorphism, we allow, before the
mapping, to perform any number of switchings on the source signed graph. This
concept has been extensively studied, and a full complexity classification
(polynomial or NP-complete) for s-homomorphism to a fixed target signed graph
has recently been obtained. Such a dichotomy is not known when we restrict the
input graph to be planar (not even for non-signed graph homomorphisms).
We show that deciding whether a (non-signed) planar graph admits a
homomorphism to the square of a cycle with , or to the circular
clique with , are NP-complete problems. We use these
results to show that deciding whether a planar signed graph admits an
s-homomorphism to an unbalanced even cycle is NP-complete. (A cycle is
unbalanced if it has an odd number of negative edges). We deduce a complete
complexity dichotomy for the planar s-homomorphism problem with any signed
cycle as a target.
We also study further restrictions involving the maximum degree and the girth
of the input signed graph. We prove that planar s-homomorphism problems to
signed cycles remain NP-complete even for inputs of maximum degree~ (except
for the case of unbalanced -cycles, for which we show this for maximum
degree~). We also show that for a given integer , the problem for signed
bipartite planar inputs of girth is either trivial or NP-complete.Comment: 17 pages, 10 figure
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