It is known that the problem of computing adjacency dimension a graph NP-hard. This suggests finding for special classes graphs or obtaining good bounds on this invariant. In work we obtain general G in terms parameters . We discuss tightness these and, some particular graphs, closed formulae. particular, show close relationships exist between and other parameters, like domination number, locat...

Felsner, Li and Trotter showed that the dimension of adjacency poset an outerplanar graph is at most 5, gave example whose has 4. We improve their upper bound to 4, which then best possible.

In this paper, we show that the dimension of the adjacency poset of a planar graph is at most 8. From below, we show that there is a planar graph whose adjacency poset has dimension 5. We then show that the dimension of the adjacency poset of an outerplanar graph is at most 5. From below, we show that there is an outerplanar graph whose adjacency poset has dimension 4. We also show that the dim...

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of knot adjacency can be used to obtain obstructions to fibering of knots and of 3-manifolds. As an application, given a fibered knot K′, we construct infinitely many non-fibered knots that share the...

Not long ago, Bagga, Beineke, and Varma [1] defined the super line multigraph of a simple graph Γ = (V,E) to be the graph Mr(Γ) whose vertex set is Pr(E), the class of r-element subsets of the edge set, and with an adjacency R ∼ R′ (where R,R′ ∈ Pr(E)) for every edge pair (e, f) with e ∈ R and f ∈ R′ such that e and f are adjacent in Γ. Thus, the number of edges joining R and R′ in Mr(Γ) is the...

In a certain class of point-line geometries, called locally projective near polygons, which includes dual polar spaces and generalised 2n-gons, surjective adjacency preserving mappings are already collineations.