Visibility Graphs, Dismantlability, and the Cops-and-Robbers Game · 2014. 5. 5. ·...

Anna LubiwUniversity of Waterloo

Visibility Graphs, Dismantlability,

and the Cops-and-Robbers Game

Visibility Graphs

Abstract

DISTANCE VISIBILITY GRAPHS

Collette CoullardDept. of Industrial Engineering and Management Sciences

Northwestern UniversityEvanston, Illinois, USAcoullard@iems. rtwu. edu

Anna LubiwDept. of Computer Science

University of WaterlooWaterloo, Ontario, Canada, N2L 3G1

alubiw@aater. waterloo, edu

A new necessary condition for a graph G to be thevisibility graph of a simple polygon is given: each 3-connected component of G must. have a vertex orderingin which every vertex is adjacent to a previous 3-clique.This property is used to give an algorithm for the dis-tance visibility graph problem: given an edge-weightedgraph G, is it the visibility graph of a simple polygonwith the given weights as Euclidean distances?

1. Introduction

Given a simple polygon P in the plane with vertex setV, the visibility graph of P, denoted G(P), is a graphon vertex set V with an edge between two vertices iffthey are visible in P—i.e. the line segment joining themremains inside the closed region of the plane boundedby P. We will insist on the non-standard qualificationthat visibility lines may not go through intermediate-rertices. The edges of P are edges of G(P). See Figure1.1.

Research of C. Coullard partially supported by theOffice of Naval Research and by the National ScienceFoundation

Research of A. Lubiw partially supported by theNatural Sciences and Engineering Research Council ofCanada

Permissmrr to copy without fee all or part ol’ this material is granted pro-vided that the copies are not made or distributed for direct commercialadvantage, the ACM copyright notice aud the title of the publication andits date appear, and notice is given that copying is by perrmssion of theAssociation for Computmg Machimxy. To copy otherwise, or torepubhsh, requires a fee and/or specific permiaaion.

Figure 1.1.

There is no polynomial time algorithm known torecognize visibility graphs—i.e. to decide given a graph,whether it is the visibility graph of some polygon. Noris the problem known to be NP-hard. III fact, it is noteven known if the problem is in NP. See O’Rourke’sbook [0’ R]. Though direct practical applications of avisibility graph recognition algorithm seem remote, weare surely missing some fundamental and potentially

~ 1991 ACM 0-89791-426-0/9 1/0006/0289 $1.50 289

Abstract

DISTANCE VISIBILITY GRAPHS

Collette CoullardDept. of Industrial Engineering and Management Sciences

Northwestern UniversityEvanston, Illinois, USAcoullard@iems. rtwu. edu

Anna LubiwDept. of Computer Science

University of WaterlooWaterloo, Ontario, Canada, N2L 3G1

alubiw@aater. waterloo, edu

A new necessary condition for a graph G to be thevisibility graph of a simple polygon is given: each 3-connected component of G must. have a vertex orderingin which every vertex is adjacent to a previous 3-clique.This property is used to give an algorithm for the dis-tance visibility graph problem: given an edge-weightedgraph G, is it the visibility graph of a simple polygonwith the given weights as Euclidean distances?

1. Introduction

Given a simple polygon P in the plane with vertex setV, the visibility graph of P, denoted G(P), is a graphon vertex set V with an edge between two vertices iffthey are visible in P—i.e. the line segment joining themremains inside the closed region of the plane boundedby P. We will insist on the non-standard qualificationthat visibility lines may not go through intermediate-rertices. The edges of P are edges of G(P). See Figure1.1.

Research of C. Coullard partially supported by theOffice of Naval Research and by the National ScienceFoundation

Research of A. Lubiw partially supported by theNatural Sciences and Engineering Research Council ofCanada

Permissmrr to copy without fee all or part ol’ this material is granted pro-vided that the copies are not made or distributed for direct commercialadvantage, the ACM copyright notice aud the title of the publication andits date appear, and notice is given that copying is by perrmssion of theAssociation for Computmg Machimxy. To copy otherwise, or torepubhsh, requires a fee and/or specific permiaaion.

Figure 1.1.

There is no polynomial time algorithm known torecognize visibility graphs—i.e. to decide given a graph,whether it is the visibility graph of some polygon. Noris the problem known to be NP-hard. III fact, it is noteven known if the problem is in NP. See O’Rourke’sbook [0’ R]. Though direct practical applications of avisibility graph recognition algorithm seem remote, weare surely missing some fundamental and potentially

~ 1991 ACM 0-89791-426-0/9 1/0006/0289 $1.50 289

polygon visibility graph

visibility graphs are a mystery with respect to other known graph classes

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JOURNAL OF ALGORITHMS 11, l-26 (1990)

Recognizing Visibility Graphs of Spiral Polygons H. EVERETT AND D. G. CORNEIL

Depurtment of Computer Science, University of Toronto, Toronto, Ontario, Canada MSS IA4

Received February 22.1988; revised September 22,198s

The recognition problem for visibility graphs is, given a graph, to determine whether it is the visibility graph of a simple polygon. The complexity of this problem is unknown. In this paper we show how to determine, in linear time, whether a graph is the visibility graph of a spiral polygon. 8 1990 Academic press. IX.

1. INTRODUCTION

A simple polygon P is a closed chain of line segments in the plane with no two non-adjacent line segments intersecting. A polygon is represented by its boundary chain, an ordered sequence, [ u$, UT,. . . , v,*- i J, of real-valued (x, -y) coordinates called the vertices of the polygon.

Two vertices ur and u,? are visible if the closed line segment between them does not intersect the exterior of P. Note that if the line segment touches the boundary of P the vertices are still considered visible. The uisibi&y graph, G, of P is the graph whose vertices correspond to the vertices of P and two vertices are adjacent in G if the corresponding vertices in P are visible. Throughout this paper the label of a vertex of P is the label of the corresponding vertex of G followed by an asterisk.

The recognition problem for visibility graphs is, given a graph, determine whether it is the visibility graph of a simple polygon. Ghosh has identified three necessary conditions for a graph to be a visibility graph and has conjectured that these are sufficient [5]. However, it is not clear that they can be checked in polynomial time. El Gindy has found a set of necessary conditions for a graph to be the visibility graph of a conuex fun [3]. El Cindy has also shown that all maximal outerplanar graphs are visibility graphs. In general, however, the recognition problem for visibility graphs is open.

1 0196-6774/90 $3.00

Copyright Cs, 1990 by Academic Press. Inc. All rights of reproduction in any form reserved.

ELSEVIER Computational Geometry 5 (1995) 51-63

Computational Geometry

Theory and Applications

Negative results on characterizing visibility graphs

H. Everett a2*, D. Corneil b a Dkpartement d’lnformatique, Uniuersit6 du Qukbec h MonhGal, Cp. 8888, WCC. A, Montrtal, Quk.,

Canada H3C 3P8 b Department of Computer Science, University of Toronto, Toronto, Ont., Canada M5S IA4

Communicated by Emo Welzl, submitted 13 December 1990; resubmitted 11 October 1994; accepted 4 November 1994

Abstract

There is no known combinatorial characterization of the visibility graphs of simple polygons. In this paper we show negative results on two different approaches to finding such a characterization. We show that Ghosh’s three necessary conditions for a graph to be a visibility graph are not sufficient thus disproving his conjecture. We also show that there is no finite set of minimal forbidden induced subgraphs that characterize visibility graphs.

1. Introduction

Visibility problems form a large class of problems in computational geometry that arise in such areas as graphics, robot motion planning, pattern recognition and VLSI design [12,14]. Some of these problems are of a combinatorial nature and visibility graphs are a combinatorial structure that capture the visibility information appropriate for their solution. For example, visibility graphs can be used to find the shortest path between two objects in the plane that avoids some set of obstacles [16,10].

Visibility graphs have been defined for several types of geometric objects; in this paper we restrict our attention to the visibility graphs of simple polygons. The visibility graph of a simple polygon is the graph whose vertices correspond to the vertices of the polygon and such that two vertices in the graph are adjacent if the corresponding polygon vertices are visible, that is, if the line segment between them does not intersect the exterior of the polygon.

?? Corresponding author.

0925-7721/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0925-7721(95)00021-2


. . . except for some special cases:

JOURNAL OF ALGORITHMS 11, l-26 (1990)

Recognizing Visibility Graphs of Spiral Polygons H. EVERETT AND D. G. CORNEIL

Depurtment of Computer Science, University of Toronto, Toronto, Ontario, Canada MSS IA4

Received February 22.1988; revised September 22,198s

The recognition problem for visibility graphs is, given a graph, to determine whether it is the visibility graph of a simple polygon. The complexity of this problem is unknown. In this paper we show how to determine, in linear time, whether a graph is the visibility graph of a spiral polygon. 8 1990 Academic press. IX.

1. INTRODUCTION

A simple polygon P is a closed chain of line segments in the plane with no two non-adjacent line segments intersecting. A polygon is represented by its boundary chain, an ordered sequence, [ u$, UT,. . . , v,*- i J, of real-valued (x, -y) coordinates called the vertices of the polygon.

Two vertices ur and u,? are visible if the closed line segment between them does not intersect the exterior of P. Note that if the line segment touches the boundary of P the vertices are still considered visible. The uisibi&y graph, G, of P is the graph whose vertices correspond to the vertices of P and two vertices are adjacent in G if the corresponding vertices in P are visible. Throughout this paper the label of a vertex of P is the label of the corresponding vertex of G followed by an asterisk.

The recognition problem for visibility graphs is, given a graph, determine whether it is the visibility graph of a simple polygon. Ghosh has identified three necessary conditions for a graph to be a visibility graph and has conjectured that these are sufficient [5]. However, it is not clear that they can be checked in polynomial time. El Gindy has found a set of necessary conditions for a graph to be the visibility graph of a conuex fun [3]. El Cindy has also shown that all maximal outerplanar graphs are visibility graphs. In general, however, the recognition problem for visibility graphs is open.

1 0196-6774/90 $3.00

Copyright Cs, 1990 by Academic Press. Inc. All rights of reproduction in any form reserved.

ELSEVIER Computational Geometry 5 (1995) 51-63

Computational Geometry

Theory and Applications

Negative results on characterizing visibility graphs

H. Everett a2*, D. Corneil b a Dkpartement d’lnformatique, Uniuersit6 du Qukbec h MonhGal, Cp. 8888, WCC. A, Montrtal, Quk.,

Canada H3C 3P8 b Department of Computer Science, University of Toronto, Toronto, Ont., Canada M5S IA4

Communicated by Emo Welzl, submitted 13 December 1990; resubmitted 11 October 1994; accepted 4 November 1994

Abstract

There is no known combinatorial characterization of the visibility graphs of simple polygons. In this paper we show negative results on two different approaches to finding such a characterization. We show that Ghosh’s three necessary conditions for a graph to be a visibility graph are not sufficient thus disproving his conjecture. We also show that there is no finite set of minimal forbidden induced subgraphs that characterize visibility graphs.

1. Introduction

Visibility problems form a large class of problems in computational geometry that arise in such areas as graphics, robot motion planning, pattern recognition and VLSI design [12,14]. Some of these problems are of a combinatorial nature and visibility graphs are a combinatorial structure that capture the visibility information appropriate for their solution. For example, visibility graphs can be used to find the shortest path between two objects in the plane that avoids some set of obstacles [16,10].

Visibility graphs have been defined for several types of geometric objects; in this paper we restrict our attention to the visibility graphs of simple polygons. The visibility graph of a simple polygon is the graph whose vertices correspond to the vertices of the polygon and such that two vertices in the graph are adjacent if the corresponding polygon vertices are visible, that is, if the line segment between them does not intersect the exterior of the polygon.

?? Corresponding author.

0925-7721/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0925-7721(95)00021-2

⊆ interval graphs


. . . except for some special cases:

Visibility Graphs

Maximal subclasses: [+]Details

SC 2-tree

maximal outerplanar

Problems

Problems in italics have no summary page and are only listed when ISGCI contains a result for the currentclass.

3-Colourability [?] Unknown to ISGCI [+]Details

Clique [?] Unknown to ISGCI [+]Details

Clique cover [?] Unknown to ISGCI [+]Details

Cliquewidth [?] Unbounded [+]Details

Cliquewidth expression [?] Unbounded or NP-complete [+]Details

Colourability [?] Unknown to ISGCI [+]Details

Cutwidth [?] Unknown to ISGCI [+]Details

Domination [?] NP-complete [+]Details

Feedback vertex set [?] Unknown to ISGCI [+]Details

Hamiltonian cycle [?] Unknown to ISGCI [+]Details

Hamiltonian path [?] Unknown to ISGCI [+]Details

ISGCI

Dismantlable Graphs

eorem. [A.L. et al.] Visibility graphs are dismantlable.

Definition. A graph G is dismantlable if it has a vertex ordering {v1,v2, . . . ,vn} such that

for each i < n, there is a vertex vj, j > i that dominates vi in the graph induced by {vi, . . . ,vn}.

Equivalently: v1 is dominated by another vertex and G − v1 is dismatlable.

Definition. u dominates v if N[u] ⊇ N[v].

v1

v3

v6

v7

v8

Dismantlable Graphseorem. [Nowakowski and Winkler, 1983. Quilliot, 1983] Graph G is dismantlable iff it is cop-win in the cops and robbers game.

Cops and Robbers Game:

coprobber



coprobber

u

v1



a cop-win graph has a vertex v1 dominated by another vertex

coprobber

u

v1

Dismantlable Graphs


Proof. Every polygon has a visibility-increasing edge: an edge (u,v) such that for every point p along the edge (u,v), V(u) ⊆V(p) ⊆V(v).

uv p

V(u)

a

b

Dismantlable Graphs



uv p

V(u)

a

b

uv p

V(u)

V(p) a

b

Dismantlable Graphs



uv p

V(u)

a

b

uv p

V(u)

V(p) a

b

uv p

V(u)

V(p)

V(v)

a

b

Dismantlable Graphs



uv p

V(u)

a

b

uv p

V(u)

V(p) a

b

uv p

V(u)

V(p)

V(v)

a

b

en v dominates u and removing u gives smaller polygon.

uv

w

Cops and Robbers in a Polygon

e cop wins the cops and robbers game on the visibility graph of a polygon. . . because the graph is dismantlable

coprobber

Cops and Robbers in a Polygon

e cop wins the cops and robbers game on the visibility graph of a polygon. . . because the graph is dismantlable

coprobber

. . . GAME OVER!

Cops and Robbers in the Interior of a Polygon

eorem. [A.L., H. Vosoughpour] e cop wins the cops and robbers game in the interior of a polygon.

coprobber



coprobber

. . . GAME OVER!



coprobber

In fact, the cop can win by playing on the reflex vertices.

Consequence: interior visibility graphs are a natural class of infinite cop-win graphs.

Geňa Hahn [Cops, Robbers and Graphs, 2007]: “As of this writing, no interesting classes of infinite cop-win graphs have been described.”



u

v

interior visibility graph



Proof. e cop wins by taking the first step of the shortest path to the robber. In particular, the cop can play on the reflex vertices. e robber is trapped in an ever-shrinking region.

the robber can’t leave Pi

ci

ci<1

li

ri<1

Pi



Proof. e cop wins by taking the first step of the shortest path to the robber. In particular, the cop can play on the reflex vertices. e robber is trapped in an ever-shrinking region.

the robber can’t leave Pi

ci

ci<1

li

ri<1

Pi

Pi+1 ⊊ Pi

ci

ci<1

li

ri

Pi

ci1

li1

Pi1r

i<1

Cops and Robbers in Curved Regions

eorem. [A.L., H. Vosoughpour] e cop wins the cops and robbers game in the interior of a closed region (with a reasonable boundary).



the cop must leave theboundary



the cop must leave theboundary r

0

c0

c1



the cop must leave theboundary r

0

c0

c1

r1

c2

r0

c0

c1



the cop must leave theboundary

the danger of going too far

r0

c0

c1

r1

c2

r0

c0

c1

Visiblity-Based Pursuit EvasionGuibas, Latombe, LaValle, Lin, Motwani. A visibility-based pursuit-evasion problem. 1999.

O(log n) pursuers suffice and are sometimes necessary.

a fast evader is caught when seen by a pursuer

474 L. J. Guibas et al.

eg Fig. 1. l-iiui examples arc shown that have similar geometry. The example in the lower right requires two pursuers, while the other three examples require only one.

For any F that has at least one hole, it is clear that at least two pursuers will be necessary; if a single pursuer is used, the evader could always move so that the hole is between the evader and pursuer. In some cases subtle changes in the geometry significantly affect H(F). Consider for example, the problems in Figure 1. Although the problems are similar, only the problem in the lower right requires two pursuers.

Consider H(F) for the case of simply-connected free spaces. Let n represent the number of edges in the free space, which is represented by a simple polygon in this case. A logarithmic worst-case bound can be established: Theorem 1 For any simply-connected free space F with n edges, H(F) = O(lgn).

Proof. The proof is built on the following observation. Suppose that two vertices of F are connected by a linear segment, thus partitioning F into two simply-connected, polygonal components, F\ and F2. If H(F\) < k and H{F2) < k for some k, then H{F) <k+\ because the same k pursuers can be used to clear both F\ and F2. This requires placing a static (k + l)th pursuer at the edge common to Fj and F2 to keep F\ cleared after the k pursuers move to F2 (assuming arbitrarily that F\ is cleared first).

In general, if two simply-connected polygonal regions share a common edge and can each be cleared by at most k pursuers, then the combined region can be cleared at most k + 1 pursuers. Recall that for any simple polygon, a pair of vertices can always be connected so that polygon is partitioned into two regions, each with at least one third of the edges of the original polygon.4 This implies that F can be recursively partitioned until a triangulation is constructed, and each triangular region only requires O(lgn) recombinations before F is obtained (i.e., the recursion depth is logarithmic in n). Based on the previous observation and the fact that each

Int.

J. C

ompu

t. G

eom

. App

l. 19

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9:47

1-49

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ww

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by 9

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9.23

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3. F

or p

erso

nal u

se o

nly.

need 2 pursuers

Pursuit Evasion versus Cops and Robbers

cops & robbers in a graph [Nowakowski & Winkler, 1983]

spac

e is d

iscre

te/

cont

inuo

usca

ptur

e by c

ontac

t/

line-

of-si

ght

full i

nfor

mati

on/

no in

form

ation

evad

er lim

ited/

unlim

ited

play

ers t

ake t

urns

/

mov

e con

tinuo

usly

visibility-based pursuit- evasion in a polygon [Guibas et al. 1999]

one edge

Pursuit EvasionGuibas, Latombe, LaValle, Lin, Motwani. A visibility-based pursuit-evasion problem. 1999.

O(log n) pursuers su!ce and are sometimes necessary.

a fast evader is caught when seen by a pursuer

474 L. J. Guibas et al.

eg Fig. 1. l-iiui examples arc shown that have similar geometry. The example in the lower right requires two pursuers, while the other three examples require only one.

For any F that has at least one hole, it is clear that at least two pursuers will be necessary; if a single pursuer is used, the evader could always move so that the hole is between the evader and pursuer. In some cases subtle changes in the geometry significantly affect H(F). Consider for example, the problems in Figure 1. Although the problems are similar, only the problem in the lower right requires two pursuers.

Consider H(F) for the case of simply-connected free spaces. Let n represent the number of edges in the free space, which is represented by a simple polygon in this case. A logarithmic worst-case bound can be established: Theorem 1 For any simply-connected free space F with n edges, H(F) = O(lgn).

Proof. The proof is built on the following observation. Suppose that two vertices of F are connected by a linear segment, thus partitioning F into two simply-connected, polygonal components, F\ and F2. If H(F\) < k and H{F2) < k for some k, then H{F) <k+\ because the same k pursuers can be used to clear both F\ and F2. This requires placing a static (k + l)th pursuer at the edge common to Fj and F2 to keep F\ cleared after the k pursuers move to F2 (assuming arbitrarily that F\ is cleared first).

In general, if two simply-connected polygonal regions share a common edge and can each be cleared by at most k pursuers, then the combined region can be cleared at most k + 1 pursuers. Recall that for any simple polygon, a pair of vertices can always be connected so that polygon is partitioned into two regions, each with at least one third of the edges of the original polygon.4 This implies that F can be recursively partitioned until a triangulation is constructed, and each triangular region only requires O(lgn) recombinations before F is obtained (i.e., the recursion depth is logarithmic in n). Based on the previous observation and the fact that each

Int.

J. C

ompu

t. G

eom

. App

l. 19

99.0

9:47

1-49

3. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific.

com

by 9

9.23

5.20

9.23

0 on

02/

04/1

3. F

or p

erso

nal u

se o

nly.

need 2 pursuers



spac

e is d

iscre

te/

cont

inuo

usca

ptur

e by c

ontac

t/

line-

of-si

ght

full i

nfor

mati

on/

no in

form

ation

evad

er lim

ited/

unlim

ited

play

ers t

ake t

urns

/

mov

e con

tinuo

usly

cops & robbers in the interior of a polygon straight line


one edge



spac

e is d

iscre

te/

cont

inuo

usca

ptur

e by c

ontac

t/

line-

of-si

ght

full i

nfor

mati

on/

no in

form

ation

evad

er lim

ited/

unlim

ited

play

ers t

ake t

urns

/

mov

e con

tinuo

usly



one edge

capturing an evader in a polygon [Bhaduaria et al. 2012] distance



spac

e is d

iscre

te/

cont

inuo

usca

ptur

e by c

ontac

t/

line-

of-si

ght

full i

nfor

mati

on/

no in

form

ation

evad

er lim

ited/

unlim

ited

play

ers t

ake t

urns

/

mov

e con

tinuo

usly



one edge

capturing an evader in a polygon [Bhaduaria et al. 2012] distance

graph searching (related to tree width) [Seymour & omas. 1993]

Conclusions

• contributions to: visibility graphs, cops and robbers, pursuit evasion

• visibility graphs ⊆ dismantlable graphs

• infinite visibility graphs are cop-win

• the cop wins the cops and robbers game in a polygon (even played in all interior points, and even for curved regions)

Conclusions

• contributions to: visibility graphs, cops and robbers, pursuit evasion

• visibility graphs ⊆ dismantlable graphs

• infinite visibility graphs are cop-win

• the cop wins the cops and robbers game in a polygon (even played in all interior points, and even for curved regions)

• OPEN. Do three cops suffice in polygonal regions with holes? (ree are sometimes necessary.)

ank you

Visibility Graphs, Dismantlability, and the Cops-and-Robbers Game · 2014. 5. 5. ·...

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