146 research outputs found
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
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The frequency assignment problem
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight.
We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR
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