137 research outputs found
The localization number and metric dimension of graphs of diameter 2
We consider the localization number and metric dimension of certain graphs of diameter , focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of , we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter and polarity graphs
The -visibility Localization Game
We study a variant of the Localization game in which the cops have limited
visibility, along with the corresponding optimization parameter, the
-visibility localization number , where is a non-negative
integer. We give bounds on -visibility localization numbers related to
domination, maximum degree, and isoperimetric inequalities. For all , we
give a family of trees with unbounded values. Extending results known
for the localization number, we show that for , every tree contains a
subdivision with . For many , we give the exact value of
for the Cartesian grid graphs, with the remaining cases
being one of two values as long as is sufficiently large. These examples
also illustrate that for all distinct choices of and
$j.
Cops and Robber -- When Capturing is not Surrounding
We consider "surrounding" versions of the classic Cops and Robber game. The
game is played on a connected graph in which two players, one controlling a
number of cops and the other controlling a robber, take alternating turns. In a
turn, each player may move each of their pieces: The robber always moves
between adjacent vertices. Regarding the moves of the cops we distinguish four
versions that differ in whether the cops are on the vertices or the edges of
the graph and whether the robber may move on/through them. The goal of the cops
is to surround the robber, i.e., occupying all neighbors (vertex version) or
incident edges (edge version) of the robber's current vertex. In contrast, the
robber tries to avoid being surrounded indefinitely. Given a graph, the
so-called cop number denotes the minimum number of cops required to eventually
surround the robber. We relate the different cop numbers of these versions and
prove that none of them is bounded by a function of the classical cop number
and the maximum degree of the graph, thereby refuting a conjecture by Crytser,
Komarov and Mackey [Graphs and Combinatorics, 2020]
Confining the robber on cographs
In a game of Cops and Robbers on graphs, usually the cops' objective is to capture the robber---a situation which the robber wants to avoid invariably. In this paper, we begin with introducing the notions of trapping and confining the robber and discussing their relations with capturing the robber. Our goal is to study the confinement of the robber on graphs that are free of a fixed path as an induced subgraph. We present some necessary conditions for graphs not containing the path on vertices (referred to as -free graphs) for some , so that cops do not have a strategy to capture or confine the robber on (Propositions 2.1, 2.3). We then show that for planar cographs and planar -free graphs the confining cop number is at most one and two, respectively (Corollary 2.4). We also show that the number of vertices of a connected cograph on which one cop does not have a strategy to confine the robber has a tight lower bound of eight. Moreover, we explore the effects of twin operations---which are well known to provide a characterization of cographs---on the number of cops required to capture or confine the robber on cographs. Finally, we pose two conjectures on confining the robber on -free graphs and the smallest planar graph of confining cop number of three
Isometric path complexity of graphs
A set of isometric paths of a graph is "-rooted", where is a
vertex of , if is one of the end-vertices of all the isometric paths in
. The isometric path complexity of a graph , denoted by , is the
minimum integer such that there exists a vertex satisfying the
following property: the vertices of any isometric path of can be
covered by many -rooted isometric paths.
First, we provide an -time algorithm to compute the isometric path
complexity of a graph with vertices and edges. Then we show that the
isometric path complexity remains bounded for graphs in three seemingly
unrelated graph classes, namely, hyperbolic graphs, (theta, prism,
pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively
studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free
graphs are extensively studied in Structural Graph Theory, e.g. in the context
of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied
in Geometric Graph Theory and Computational Geometry. Our results also show
that the distance functions of these (structurally) different graph classes are
more similar than previously thought.
There is a direct algorithmic consequence of having small isometric path
complexity. Specifically, we show that if the isometric path complexity of a
graph is bounded by a constant, then there exists a polynomial-time
constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose
objective is to cover all vertices of a graph with a minimum number of
isometric paths. This applies to all the above graph classes.Comment: A preliminary version appeared in the proceedings of the MFCS 2023
conferenc
Graphs with Large Girth and Small Cop Number
In this paper we consider the cop number of graphs with no, or few, short
cycles. We show that when the girth of is at least and the minimum
degree is sufficiently large, where
, then as where . This extends
work of Frankl and implies that if is large and dense in the sense that
while also having girth , then
satisfies Meyniel's conjecture, that is . Moreover, it
implies that if is large and dense in the sense that there for some , while also having girth , then
there exists an such that , thereby
satisfying the weak Meyniel's conjecture. Of course, this implies similar
results for dense graphs with small, that is , numbers of
short cycles, as each cycle can be broken by adding a single cop. We also, show
that there are graphs with girth and minimum degree such that
the cop number is at most . This
resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by
showing that the constant cannot be improved in the exponent of a
lower bound .Comment: 7 pages, 0 figures, 0 table
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Parameterized Analysis of the Cops and Robber Game
Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber (CnR) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber. The aim is to compute the cop number of a graph, k, which is the minimum number of cops that ensures the capture of the robber.
From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number (vcn). First, we establish that k ? vcn/3+1. Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number, and design a linear vertex kernel for neighborhood diversity. Additionally, we extend all of our results to several well-studied variations of CnR
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