10 research outputs found
Making Octants Colorful and Related Covering Decomposition Problems
We give new positive results on the long-standing open problem of geometric
covering decomposition for homothetic polygons. In particular, we prove that
for any positive integer k, every finite set of points in R^3 can be colored
with k colors so that every translate of the negative octant containing at
least k^6 points contains at least one of each color. The best previously known
bound was doubly exponential in k. This yields, among other corollaries, the
first polynomial bound for the decomposability of multiple coverings by
homothetic triangles. We also investigate related decomposition problems
involving intervals appearing on a line. We prove that no algorithm can
dynamically maintain a decomposition of a multiple covering by intervals under
insertion of new intervals, even in a semi-online model, in which some coloring
decisions can be delayed. This implies that a wide range of sweeping plane
algorithms cannot guarantee any bound even for special cases of the octant
problem.Comment: version after revision process; minor changes in the expositio
Conflict Free Connectivity and the Conflict-Free-Connection Number of Graphs
We explore a relatively new concept in edge-colored graphs called conflict-free connectivity. A conflict-free path is a (edge-) colored path that has an edge with a color that appears only once. Conflict-free connectivity is the maximal number of internally disjoint conflict-free paths between all pairs of vertices in a graph. We also define the c-conflict-free-connection of a graph G. This is the maximum conflict-free connectivity of G over all c-colorings of the edges of G. In this paper we will briefly survey the works related to conflict-free connectivity. In addition, we will use the probabilistic method to achieve a bound on the c-conflict-free connection number of complete graphs
Minimal dominating sets enumeration with FPT-delay parameterized by the degeneracy and maximum degree
At STOC 2002, Eiter, Gottlob, and Makino presented a technique called ordered
generation that yields an -delay algorithm listing all minimal
transversals of an -vertex hypergraph of degeneracy . Recently at IWOCA
2019, Conte, Kant\'e, Marino, and Uno asked whether this XP-delay algorithm
parameterized by could be made FPT-delay parameterized by and the
maximum degree , i.e., an algorithm with delay for some computable function . Moreover, as a first step toward
answering that question, they note that the same delay is open for the
intimately related problem of listing all minimal dominating sets in graphs. In
this paper, we answer the latter question in the affirmative.Comment: 18 pages, 2 figure
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
The potential to improve the choice: list conflict-free coloring for geometric hypergraphs
Given a geometric hypergraph (or a range-space) , a coloring of its vertices is said to be conflict-free if for every hyperedge there is at least one vertex in whose color is distinct from the colors of all other vertices in . The study of this notion is motivated by frequency assignment problems in wireless networks. We study the list-coloring (or choice) version of this notion. In this version, each vertex is associated with a set of (admissible) colors and it is allowed to be colored only with colors from its set. List coloring arises naturally in the context of wireless networks. Our main result is a list coloring algorithm based on a new potential method. The algorithm produces a stronger unique-maximum coloring, in which colors are positive integers and the maximum color in every hyperedge occurs uniquely. As a corollary, we provide asymptotically sharp bounds on the size of the lists required to assure the existence of such unique-maximum colorings for many geometric hypergraphs (e.g., discs or pseudo-discs in the plane or points with respect to discs). Moreover, we provide an algorithm, such that, given a family of lists with the appropriate sizes, computes such a coloring from these lists
Competitive versions of vertex ranking and game acquisition, and a problem on proper colorings
In this thesis we study certain functions on graphs. Chapters 2 and 3 deal with variations on vertex ranking, a type of node-labeling scheme that models a parallel processing problem. A k-ranking of a graph G is a labeling of its vertices from {1,...,k} such that any nontrivial path whose endpoints have the same label contains a vertex with a larger label. In Chapter 2, we investigate the problem of list ranking, wherein every vertex of G is assigned a set of possible labels, and a ranking must be constructed by labeling each vertex from its list; the list ranking number of G is the minimum k such that if every vertex is assigned a set of k possible labels, then G is guaranteed to have a ranking from these lists. We compute the list ranking numbers of paths, cycles, and trees with many leaves. In Chapter 3, we investigate the problem of on-line ranking, which asks for an algorithm to rank the vertices of G as they are revealed one at a time in the subgraph of G induced by the vertices revealed so far. The on-line ranking number of G is the minimum over all such labeling algorithms of the largest label that the algorithm can be forced to use. We give algorithmic bounds on the on-line ranking number of trees in terms of maximum degree, diameter, and number of internal vertices.
Chapter 4 is concerned with the connectedness and Hamiltonicity of the graph G^j_k(H), whose vertices are the proper k-colorings of a given graph H, with edges joining colorings that differ only on a set of vertices contained within a connected subgraph of H on at most j vertices. We introduce and study the parameters g_k(H) and h_k(H), which denote the minimum j such that G^j_k(H) is connected or Hamiltonian, respectively. Finally, in Chapter 5 we compute the game acquisition number of complete bipartite graphs. An acquisition move in a weighted graph G consists a vertex v taking all the weight from a neighbor whose weight is at most the weight of v. In the acquisition game on G, each vertex initially has weight 1, and players Min and Max alternate acquisition moves until the set of vertices remaining with positive weight is an independent set. Min seeks to minimize the size of the final independent set, while Max seeks to maximize it; the game acquisition number is the size of the final set under optimal play