Abstract
We study contact representations of edge-weighted planar graphs, where vertices are rectangles or rectilinear polygons and edges are polygon contacts whose lengths represent the edge weights. We show that for any given edge-weighted planar graph whose outer face is a quadrangle, that is internally triangulated and that has no separating triangles we can construct in linear time an edge-proportional rectangular dual if one exists and report failure otherwise. For a given combinatorial structure of the contact representation and edge weights interpreted as lower bounds on the contact lengths, a corresponding contact representation that minimizes the size of the enclosing rectangle can be found in linear time. If the combinatorial structure is not fixed, we prove NP-hardness of deciding whether a contact representation with bounded contact lengths exists. Finally, we give a complete characterization of the rectilinear polygon complexity required for representing biconnected internally triangulated graphs: For outerplanar graphs complexity 8 is sufficient and necessary, and for graphs with two adjacent or multiple non-adjacent internal vertices the complexity is unbounded.
Highlights
Representing graphs by intersections or contacts of geometric objects has a long history in graph theory and graph drawing, which is covered in monographs and surveys [13,21]
We present a linear-time algorithm that decides whether a given graph G has an edge-proportional rectangular dual (EPRD) with four outer rectangles and constructs it in the positive case
We show that for any biconnected outerplanar graph G = (V, E) with weight function ω and a reference edge e ∈ E on the outer face, there exists an edge-proportional rectilinear representation P such that for each edge uv on the outer face with uv = e, there exists a U-shape whose left and right boundary are formed by the polygons P(u) and P(v), whose open side points to the top, and whose width is at most ε/2, where ε is the smallest weight of all edges
Summary
Representing graphs by intersections or contacts of geometric objects has a long history in graph theory and graph drawing, which is covered in monographs and surveys [13,21]. Rectangular duals and rectilinear representations with low-complexity polygons have practical applications, e.g., in VLSI design, cartography, or floor planning and surveillance in buildings [22] In these applications, the area of vertex polygons and/or the boundary length of adjacent polygons often play an important role, e.g., in building surveillance polygon area is linked to the number of persons in a room and boundary length represents the number of transitions from one room to the next. We present a linear-time algorithm that decides whether a given graph G has an edge-proportional rectangular dual (EPRD) with four outer rectangles and constructs it in the positive case. On the other hand, the graph has two adjacent or multiple non-adjacent internal vertices, polygons of unbounded complexity are sometimes necessary This completely characterizes the complexity of edge-proportional rectilinear representations for internally triangulated graphs
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