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Abstract. A bar 1-visibility drawing of a graph G is a drawing of G where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must connect the horizontal line segments representing the end vertices and a vertical line segment corresponding to an edge intersects at most one bar which is not an end point of the edge. A graph G is bar 1-visible if G has a bar 1-visibility drawing. A graph G is 1-planar if G has a drawing in a 2-dimensional plane such that an edge crosses at most one other edge. In this paper we give linear-time algorithms to find bar 1-visibility drawings of diagonal grid graphs and maximal outer 1-planar graphs. We also show that recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs are bar 1-visible graphs.

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Let S = {p1, p2,. .. , pn} be a set of pairwise disjoint geometric objects of some type in a 2D plane and let C = {c1, c2,. .. , cn} be a set of closed objects of some type in the same plane with the property that each element in C covers exactly one element in S and any two elements in C are interior-disjoint. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) has a vertex for each element of C and an edge between two vertices whenever the corresponding cover elements touch. It is known how to construct, for any given point seed set, a disk or triangle cover whose contact graph is 1or 2-connected but the problem of deciding whether a k-connected CCG can be constructed or not for k > 2 is still unsolved. A triangle cover contact graph (T CCG) is a cover contact graph whose cover elements are triangles. In this paper, we give algorithms to construct a 3-connected T CCG and a 4-connected T CCG for a given set of point seeds. We also show that any connected outerplanar graph has a realization as a T CCG on a given set of collinear point seeds. Note that, under this restriction, only trees and cycles are known to be realizable as CCG.

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