17,956 research outputs found
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Acyclicity in edge-colored graphs
A walk in edge-colored graphs is called properly colored (PC) if every
pair of consecutive edges in is of different color. We introduce and study
five types of PC acyclicity in edge-colored graphs such that graphs of PC
acyclicity of type is a proper superset of graphs of acyclicity of type
, The first three types are equivalent to the absence of PC
cycles, PC trails, and PC walks, respectively. While graphs of types 1, 2 and 3
can be recognized in polynomial time, the problem of recognizing graphs of type
4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when
only two colors are used). The same problem with respect to type 5 is
polynomial-time solvable for all edge-colored graphs. Using the five types, we
investigate the border between intractability and tractability for the problems
of finding the maximum number of internally vertex disjoint PC paths between
two vertices and the minimum number of vertices to meet all PC paths between
two vertices
Partitioning 2-edge-colored graphs by monochromatic paths and cycles
We present results on partitioning the vertices of -edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every -edge-colored graph
can be partitioned into at most monochromatic cycles, where
denotes the independence number of . Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from vertices, the vertex
set of any -edge-colored graph with minimum degree at least
(1+\eps){3|V(G)|\over 4} can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that does not contain a fixed bipartite graph , we show that
in every -edge-coloring of , vertices can be covered by two
vertex disjoint paths of different colors, where is a constant depending
only on . In particular, we prove that , which is best possible
Observable Graphs
An edge-colored directed graph is \emph{observable} if an agent that moves
along its edges is able to determine his position in the graph after a
sufficiently long observation of the edge colors. When the agent is able to
determine his position only from time to time, the graph is said to be
\emph{partly observable}. Observability in graphs is desirable in situations
where autonomous agents are moving on a network and one wants to localize them
(or the agent wants to localize himself) with limited information. In this
paper, we completely characterize observable and partly observable graphs and
show how these concepts relate to observable discrete event systems and to
local automata. Based on these characterizations, we provide polynomial time
algorithms to decide observability, to decide partial observability, and to
compute the minimal number of observations necessary for finding the position
of an agent. In particular we prove that in the worst case this minimal number
of observations increases quadratically with the number of nodes in the graph.
From this it follows that it may be necessary for an agent to pass through
the same node several times before he is finally able to determine his position
in the graph. We then consider the more difficult question of assigning colors
to a graph so as to make it observable and we prove that two different versions
of this problem are NP-complete.Comment: 15 pages, 8 figure
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