ISSN:
1439-6912
Keywords:
05C
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Colin de Vedière introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G)≤2 if and only ifG is outerplanar, and μ(G)≤3 if and only ifG is planar. We prove that if the complement of a graphG onn nodes is outerplanar, then μ(G)≥n−4, and if it is planar, then μ(G)≥n−5. We give a full characterization of maximal planar graphs whose complementsG have μ(G)=n−5. In the opposite direction we show that ifG does not have “twin” nodes, then μ(G)≥n−3 implies that the complement ofG is outerplanar, and μ(G)≥n−4 implies that the complement ofG is planar. Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01195002
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