POINT VISIBILITY GRAPHS AND ${\mathcal O}$-CONVEX COVER

2000 ◽  
Vol 10 (01) ◽  
pp. 55-71
Author(s):  
DAVID BREMNER ◽  
THOMAS SHERMER
Keyword(s):  
Polynomial Algorithm ◽  
Local Maximum ◽  
Visibility Graph ◽  
Perfect Graphs ◽  
Visibility Graphs ◽  
Point Set ◽  
Convex Cover ◽  
Visibility Relation ◽  
Point Visibility

A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subproblems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, under a generalized notion of visibility called [Formula: see text]-visibility. Let [Formula: see text] denote a set of line orientations. A connected point set P is called [Formula: see text]-convex if the intersection of P with any line with orientation in [Formula: see text] is connected. Two points in a polygon are said to be [Formula: see text]-visible if there is an [Formula: see text]-convex path between them inside the polygon. Let [Formula: see text] denote the set of orientations perpendicular to orientations in [Formula: see text]. Let [Formula: see text] be the set of orientations θ such that a "reflex" local maximum in the boundary of P exists with respect to θ. Our characterization of which polygons have weakly-triangulated PVGs is based on restricting the cardinality and span of [Formula: see text]. This characterization allows us to exhibit a class of polygons admitting a polynomial algorithm for [Formula: see text]-convex cover.

scholarly journals Point Visibility Graph Recognition is NP-Hard

2016 ◽  
Vol 26 (01) ◽  
pp. 1-32 ◽  
Author(s):  
Bodhayan Roy
Keyword(s):  
Polynomial Time ◽  
Visibility Graph ◽  
Np Hard ◽  
Graph Recognition ◽  
Visibility Graphs ◽  
Point Visibility

Given a 3-SAT formula, a graph can be constructed in polynomial time such that the graph is a point visibility graph if and only if the 3-SAT formula is satisfiable. This reduction establishes that the problem of recognition of point visibility graphs is NP-hard.

scholarly journals Topological Approach to Mathematical Programs with Switching Constraints

2021 ◽  
Author(s):  
Vladimir Shikhman
Keyword(s):  
Stationary Point ◽  
Mountain Pass Theorem ◽  
Strong Stability ◽  
Complete Characterization ◽  
Stationary Points ◽  
Mathematical Programs ◽  
Linear Independence Constraint Qualification ◽  
Point Set ◽  
Stationary Level

AbstractWe study mathematical programs with switching constraints (for short, MPSC) from the topological perspective. Two basic theorems from Morse theory are proved. Outside the W-stationary point set, continuous deformation of lower level sets can be performed. However, when passing a W-stationary level, the topology of the lower level set changes via the attachment of a w-dimensional cell. The dimension w equals the W-index of the nondegenerate W-stationary point. The W-index depends on both the number of negative eigenvalues of the restricted Lagrangian’s Hessian and the number of bi-active switching constraints. As a consequence, we show the mountain pass theorem for MPSC. Additionally, we address the question if the assumption on the nondegeneracy of W-stationary points is too restrictive in the context of MPSC. It turns out that all W-stationary points are generically nondegenerate. Besides, we examine the gap between nondegeneracy and strong stability of W-stationary points. A complete characterization of strong stability for W-stationary points by means of first and second order information of the MPSC defining functions under linear independence constraint qualification is provided. In particular, no bi-active Lagrange multipliers of a strongly stable W-stationary point can vanish.

scholarly journals A new characterization of perfect graphs

2012 ◽  
Vol 312 (17) ◽  
pp. 2751-2755 ◽  
Author(s):  
Hortensia Galeana-Sánchez
Keyword(s):  
Perfect Graphs

scholarly journals An induced subgraph characterization of domination perfect graphs

1995 ◽  
Vol 20 (3) ◽  
pp. 375-395 ◽  
Author(s):  
Igor E. Zvervich ◽  
Vadim E. Zverovich
Keyword(s):  
Perfect Graphs ◽  
Induced Subgraph

scholarly journals A Characterization of b-Perfect Graphs

2011 ◽  
Vol 71 (1) ◽  
pp. 95-122 ◽  
Author(s):  
Chính T. Hoàng ◽  
Frédéric Maffray ◽  
Meriem Mechebbek
Keyword(s):  
Perfect Graphs

scholarly journals Semiarcs with Long Secants

10.37236/3771 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Bence Csajbók
Keyword(s):  
Lower Bound ◽  
Projective Plane ◽  
Complete Characterization ◽  
Symmetric Difference ◽  
Blocking Set ◽  
Point Set

In a projective plane $\Pi_q$ of order $q$, a non-empty point set $\mathcal{S}_t$ is a $t$-semiarc if the number of tangent lines to $\mathcal{S}_t$ at each of its points is $t$. If $\mathcal{S}_t$ is a $t$-semiarc in $\Pi_q$, $t<q$, then each line intersects $\mathcal{S}_t$ in at most $q+1-t$ points. Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in $\Pi_q$ only if $q\leq 3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\geq(q-1)/2$, with equality only if $\mathcal{S}_t$ is a blocking set of Rédei type of size $3(q+1)/2$.We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$.

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