1,049 research outputs found
Optimizing the trade-off between number of cops and capture time in Cops and Robbers
The cop throttling number of a graph for the game of Cops and
Robbers is the minimum of , where is the number of cops and
is the minimum number of rounds needed for cops to capture the
robber on over all possible games in which both players play optimally. In
this paper, we construct a family of graphs having ,
establish a sublinear upper bound on the cop throttling number, and show that
the cop throttling number of chordal graphs is . We also introduce
the product cop throttling number as a parameter that
minimizes the person-hours used by the cops. This parameter extends the notion
of speed-up that has been studied in the context of parallel processing and
network decontamination. We establish bounds on the product cop throttling
number in terms of the cop throttling number, characterize graphs with low
product cop throttling number, and show that for a chordal graph ,
.Comment: 19 pages, 3 figure
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
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