Let G be a graph with vertex set V. Let F be a field, and f a function from V to Fk such that xy is an edge of G if and only if f(x)·f(y) ≥ t. This is the dot product representation of G. The number t is called the dot product threshold, and the smallest possible value of k is called the dot product dimension.
Fiduccia, Scheinerman, Trenk and Zito (1998) asked whether every planar graph is a 3-dot product graph. We show that the answer is “no”. On the other hand, ...
Dot product representations of graphs - ScienceDirect.com
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We call a dot product representation of a graph rational provided all coordinates of all the vectors in the representation are rational numbers. Page 3. C.M. ...
Oct 23, 2010 · We study a type of geometric representation of graphs using vectors from Rk for some k ∈ N. Let G be a graph. We say G is a k-dot product ...
Nov 7, 2011 · Abstract. A graph G G is a k k -dot product graph if there exists a vector labelling u:V(G)→Rk u : V ( G ) → R k such that u(i)Tu(j)≥1 u ...
Fiduccia, Scheinerman, Trenk and Zito (1998) asked whether every planar graph is a 3-dot product graph. We show that the answer is “no”. On the other hand, ...
Fiduccia et al. determined the dot product dimension of complete multipartite, bipartite, interval graphs and trees [3]. An outer planar graph is a graph that ...
A graph G is a k-dot product graph if there exists a vector labelling u:V(G)→ℝ k such that u(i) T u(j)≥1 if and only if ij∈E(G). C. M. Fiduccia et al.
ij ∈ E(G). An explicit set of vectors in Rk that exhibits G in this way is called a k-dot product representation of G. The dot product dimension of G is ...
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This paper shows that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and each face, so ...
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