The Partial Visibility Representation Extension Problem

For a graph G , a function ψ is called a bar visibility representation of G when for each vertex v ∈ V ( G ) , ψ ( v ) is a horizontal line segment ( bar ) and u v ∈ E ( G ) if and only if there is an unobstructed, vertical, ε -wide line of sight between ψ ( u ) and ψ ( v ) . Graphs admitting such r...

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Published in:	Algorithmica Vol. 80; no. 8; pp. 2286 - 2323
Main Authors:	Chaplick, Steven, Guśpiel, Grzegorz, Gutowski, Grzegorz, Krawczyk, Tomasz, Liotta, Giuseppe
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.08.2018 Springer Nature B.V
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphical representations Graphs Mathematics of Computing Set theory Theory of Computation Visibility Bar visibility SPQR-trees St-ordering Partial representation extension Planar graphs NP-completeness
ISSN:	0178-4617, 1432-0541
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Abstract	For a graph G , a function ψ is called a bar visibility representation of G when for each vertex v ∈ V ( G ) , ψ ( v ) is a horizontal line segment ( bar ) and u v ∈ E ( G ) if and only if there is an unobstructed, vertical, ε -wide line of sight between ψ ( u ) and ψ ( v ) . Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G , a bar visibility representation of G , additionally, puts the bar ψ ( u ) strictly below the bar ψ ( v ) for each directed edge ( u , v ) of G . We study a generalization of the recognition problem where a function ψ ′ defined on a subset V ′ of V ( G ) is given and the question is whether there is a bar visibility representation ψ of G with ψ ( v ) = ψ ′ ( v ) for every v ∈ V ′ . We show that for undirected graphs this problem, and other closely related problems, is NP -complete, but for certain cases involving directed graphs it is solvable in polynomial time.
AbstractList	For a graph G, a function ψ is called a bar visibility representation of G when for each vertex v∈V(G), ψ(v) is a horizontal line segment (bar) and uv∈E(G) if and only if there is an unobstructed, vertical, ε-wide line of sight between ψ(u) and ψ(v). Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G, a bar visibility representation of G, additionally, puts the bar ψ(u) strictly below the bar ψ(v) for each directed edge (u, v) of G. We study a generalization of the recognition problem where a function ψ′ defined on a subset V′ of V(G) is given and the question is whether there is a bar visibility representation ψ of G with ψ(v)=ψ′(v) for every v∈V′. We show that for undirected graphs this problem, and other closely related problems, is NP-complete, but for certain cases involving directed graphs it is solvable in polynomial time. For a graph G , a function ψ is called a bar visibility representation of G when for each vertex v ∈ V ( G ) , ψ ( v ) is a horizontal line segment ( bar ) and u v ∈ E ( G ) if and only if there is an unobstructed, vertical, ε -wide line of sight between ψ ( u ) and ψ ( v ) . Graphs admitting such representations are well understood (via simple characterizations) and recognizable in linear time. For a directed graph G , a bar visibility representation of G , additionally, puts the bar ψ ( u ) strictly below the bar ψ ( v ) for each directed edge ( u , v ) of G . We study a generalization of the recognition problem where a function ψ ′ defined on a subset V ′ of V ( G ) is given and the question is whether there is a bar visibility representation ψ of G with ψ ( v ) = ψ ′ ( v ) for every v ∈ V ′ . We show that for undirected graphs this problem, and other closely related problems, is NP -complete, but for certain cases involving directed graphs it is solvable in polynomial time.
Author	Gutowski, Grzegorz Guśpiel, Grzegorz Chaplick, Steven Krawczyk, Tomasz Liotta, Giuseppe
Author_xml	– sequence: 1 givenname: Steven orcidid: 0000-0003-3501-4608 surname: Chaplick fullname: Chaplick, Steven organization: Lehrstuhl für Informatik I, Universität Würzburg – sequence: 2 givenname: Grzegorz orcidid: 0000-0002-3303-8107 surname: Guśpiel fullname: Guśpiel, Grzegorz organization: Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University – sequence: 3 givenname: Grzegorz orcidid: 0000-0003-3313-1237 surname: Gutowski fullname: Gutowski, Grzegorz email: gutowski@tcs.uj.edu.pl organization: Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University – sequence: 4 givenname: Tomasz surname: Krawczyk fullname: Krawczyk, Tomasz organization: Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University – sequence: 5 givenname: Giuseppe surname: Liotta fullname: Liotta, Giuseppe organization: Universitá degli Studi di Perugia
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Snippet	For a graph G , a function ψ is called a bar visibility representation of G when for each vertex v ∈ V ( G ) , ψ ( v ) is a horizontal line segment ( bar ) and... For a graph G, a function ψ is called a bar visibility representation of G when for each vertex v∈V(G), ψ(v) is a horizontal line segment (bar) and uv∈E(G) if...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphical representations Graphs Mathematics of Computing Set theory Theory of Computation Visibility
Title	The Partial Visibility Representation Extension Problem
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