Angular resolution (graph drawing)

This drawing of a hypercube graph has angular resolution Template:Math.

In graph drawing, the angular resolution of a drawing of a graph is the sharpest angle formed by any two edges that meet at a common vertex of the drawing.

Properties

Relation to vertex degree

Template:Harvtxt observed that every straight-line drawing of a graph with maximum degree Template:Mvar has angular resolution at most Template:Math: if Template:Mvar is a vertex of degree Template:Mvar, then the edges incident to Template:Mvar partition the space around Template:Mvar into Template:Mvar wedges with total angle Template:Math, and the smallest of these wedges must have an angle of at most Template:Math. More strongly, if a graph is Template:Mvar-regular, it must have angular resolution less than $\frac{\pi }{d-1}$, because this is the best resolution that can be achieved for a vertex on the convex hull of the drawing.

Relation to graph coloring

As Template:Harvtxt showed, the largest possible angular resolution of a graph Template:Mvar is closely related to the chromatic number of the square Template:Math, the graph on the same vertex set in which pairs of vertices are connected by an edge whenever their distance in Template:Mvar is at most two. If Template:Math can be colored with Template:Math colors, then G may be drawn with angular resolution Template:Math, for any Template:Math, by assigning distinct colors to the vertices of a [[regular polygon|regular Template:Math-gon]] and placing each vertex of Template:Mvar close to the polygon vertex with the same color. Using this construction, they showed that every graph with maximum degree Template:Mvar has a drawing with angular resolution proportional to Template:Math. This bound is close to tight: they used the probabilistic method to prove the existence of graphs with maximum degree Template:Mvar whose drawings all have angular resolution $O\left(\frac{\log d}{d^2}\right)$.

Existence of optimal drawings

Template:Harvtxt provided an example showing that there exist graphs that do not have a drawing achieving the maximum possible angular resolution; instead, these graphs have a family of drawings whose angular resolutions tend towards some limiting value without reaching it. Specifically, they exhibited an 11-vertex graph that has drawings of angular resolution Template:Math for any Template:Math, but that does not have a drawing of angular resolution exactly Template:Math.

Special classes of graphs

Trees

Every tree may be drawn in such a way that the edges are equally spaced around each vertex, a property known as perfect angular resolution. Moreover, if the edges may be freely permuted around each vertex, then such a drawing is possible, without crossings, with all edges unit length or higher, and with the entire drawing fitting within a bounding box of polynomial area. However, if the cyclic ordering of the edges around each vertex is already determined as part of the input to the problem, then achieving perfect angular resolution with no crossings may sometimes require exponential area.

Outerplanar graphs

Perfect angular resolution is not always possible for outerplanar graphs, because vertices on the convex hull of the drawing with degree greater than one cannot have their incident edges equally spaced around them. Nonetheless, every outerplanar graph of maximum degree Template:Mvar has an outerplanar drawing with angular resolution proportional to Template:Math.

Planar graphs

For planar graphs with maximum degree Template:Mvar, the square-coloring technique of Template:Harvtxt provides a drawing with angular resolution proportional to Template:Math, because the square of a planar graph must have chromatic number proportional to Template:Mvar. More precisely, Wegner conjectured in 1977 that the chromatic number of the square of a planar graph is at most $\max (d+5,\frac{3d}{2}+1)$, and it is known that the chromatic number is at most $\frac{5d}{3}+O(1)$. However, the drawings resulting from this technique are generally not planar.

For some planar graphs, the optimal angular resolution of a planar straight-line drawing is Template:Math, where Template:Mvar is the degree of the graph. Additionally, such a drawing may be forced to use very long edges, longer by an exponential factor than the shortest edges in the drawing. Template:Harvtxt used the circle packing theorem and ring lemma to show that every planar graph with maximum degree Template:Mvar has a planar drawing whose angular resolution is at worst an exponential function of Template:Mvar, independent of the number of vertices in the graph.

Computational complexity

It is NP-hard to determine whether a given graph of maximum degree Template:Mvar has a drawing with angular resolution Template:Math, even in the special case that Template:Math. However, for certain restricted classes of drawings, including drawings of trees in which extending the leaves to infinity produces a convex subdivision of the plane as well as drawings of planar graphs in which each bounded face is a centrally-symmetric polygon, a drawing of optimal angular resolution may be found in polynomial time.

History

Angular resolution was first defined by Template:Harvtxt.

Although originally defined only for straight-line drawings of graphs, later authors have also investigated the angular resolution of drawings in which the edges are polygonal chains, circular arcs, or spline curves.

The angular resolution of a graph is closely related to its crossing resolution, the angle formed by crossings in a drawing of the graph. In particular, RAC drawing seeks to ensure that these angles are all right angles, the largest crossing angle possible.Template:Sfnp

References

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