114 research outputs found
Backbone colorings for networks: tree and path backbones
We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph and a spanning subgraph of (the backbone of ), a backbone coloring for and is a proper vertex coloring of in which the colors assigned to adjacent vertices in differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path
Sparse square roots.
We show that it can be decided in polynomial time whether a graph of maximum degree 6 has a square root; if a square root exists, then our algorithm finds one with minimum number of edges. We also show that it is FPT to decide whether a connected n-vertex graph has a square root with at most nâââ1â+âk edges when this problem is parameterized by k. Finally, we give an exact exponential time algorithm for the problem of finding a square root with maximum number of edges
Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.
We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time
Induced disjoint paths in circular-arc graphs in linear time
The Induced Disjoint Paths problem is to test whether an graph G on n vertices with k distinct pairs of vertices (si,ti) contains paths P1,âŠ,Pk such that Pi connects si and ti for i=1,âŠ,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1â€
Graph editing to a given degree sequence
We investigate the parameterized complexity of the graph editing problem called Editing to a Graph with a Given Degree Sequence where the aim is to obtain a graph with a given degree sequence ÏÏ by at most k vertex or edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2O(k(Î+k)2)n2logn2O(k(Î+k)2)n2logâĄn for n-vertex graphs, where Î=maxÏÎ=maxÏ, i.e., the problem is FPT when parameterized by k+Îk+Î. We also show that Editing to a Graph with a Given Degree Sequence has a polynomial kernel when parameterized by k+Îk+Î if only edge additions are allowed, and there is no polynomial kernel unless NPâcoNP/polyNPâcoNP/poly for all other combinations of allowed editing operations
Parametrized Complexity of Weak Odd Domination Problems
Given a graph , a subset of vertices is a weak odd
dominated (WOD) set if there exists such that
every vertex in has an odd number of neighbours in . denotes
the size of the largest WOD set, and the size of the smallest
non-WOD set. The maximum of and , denoted
, plays a crucial role in quantum cryptography. In particular
deciding, given a graph and , whether is of
practical interest in the design of graph-based quantum secret sharing schemes.
The decision problems associated with the quantities , and
are known to be NP-Complete. In this paper, we consider the
approximation of these quantities and the parameterized complexity of the
corresponding problems. We mainly prove the fixed-parameter intractability
(W-hardness) of these problems. Regarding the approximation, we show that
, and admit a constant factor approximation
algorithm, and that and have no polynomial approximation
scheme unless P=NP.Comment: 16 pages, 5 figure
List Coloring in the Absence of Two Subgraphs
list assignment of a graph G = (V;E) is a function L that assigns a list L(u) of so-called admissible colors to each u 2 V . The List Coloring problem is that of testing whether a given graph G = (V;E) has a coloring c that respects a given list assignment L, i.e., whether G has a mapping c : V ! f1; 2; : : :g such that (i) c(u) 6= c(v) whenever uv 2 E and (ii) c(u) 2 L(u) for all u 2 V . If a graph G has no induced subgraph isomorphic to some graph of a pair fH1;H2g, then G is called (H1;H2)-free. We completely characterize the complexity of List Coloring for (H1;H2)-free graphs
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