Some Results On Convex Greedy Embedding Conjecture for -Connected Planar Graphs
A greedy embedding of a graph into a metric space is a function such that in the embedding for every pair of non-adjacent vertices there exists another vertex adjacent to which is closer to than . This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every -connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all -connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a -connected planar graph , an embedding of is a planar convex greedy embedding if and only if, in the embedding , weight of the maximum weight spanning tree () and weight of the minimum weight spanning tree () satisfies , for some . In order to present this result we define a notion of weak greedy embedding. For a –weak greedy embedding of a graph is a planar embedding such that for every pair of non-adjacent vertices there exists a vertex adjacent to such that distance between and is at most times the distance between and . We show that any three connected planar graph has a –weak greedy planar convex embedding in the Euclidean plane with , where is the ratio of maximum and minimum distance between pair of vertices in the embedding of . Finally, we also show that this bound is tight for well known Tutte embedding of -connected planar graphs in the Euclidean plane - which is planar and convex.
1.1 Greedy embedding conjecture
An embedding of an undirected graph in a metric space is a mapping . In this work we will be concerned with a special case when is the plane () endowed with the Euclidean (i.e. ) metric. The function then maps each edge of the graph to the line-segments joining the images of its end points. We say that embedding is planar when no two such line-segments (edges) intersect at any point other than their end points. Let denote the Euclidean distance between two points and .
Greedy embedding (): A greedy embedding of a graph into a metric space is a function with the following property: for every pair of non-adjacent vertices there exists a vertex adjacent to such that .
This notion of greedy embedding was defined by Papadimitriou and Ratajczak in . They have presented graphs which do not admit a greedy embedding in the Euclidean plane, and conjectured following:
Conjecture 1 (Greedy embedding conjecture).
Every -connected planar graph has a greedy embedding in the Euclidean plane.
A convex embedding of a planar graph is a “planar embedding” with a property that all faces, including the external faces are “convex”. Additionally, Papadimitriou and Ratajczak stated the following stronger form of the conjecture:
Conjecture 2 (Convex greedy embedding conjecture).
Every -connected planar graph has a greedy convex embedding in the Euclidean plane.
Note that every -connected planar graph has a convex embedding in the Euclidean plane (using Tutte’s rubber band algorithm [2, 3, 4, 5]). In  it was shown that admits no greedy embedding for . Which imply that both hypotheses of the conjecture are necessary: there exist graphs that are planar but not -connected (), or -connected but not planar (), that does not admits any greedy embedding. Also, they show that high connectivity alone does not guarantee a greedy embedding. Papadimitriou and Ratajczak in  also provided examples of graphs which have a greedy embedding (e.g., Hamiltonian graphs). Note that if is a spanning subgraph of , i.e. then every greedy embedding of is also a greedy embedding of . Hence, the conjecture extends to any graph having a -connected planar spanning subgraph.
1.2 Known results
Recently, greedy embedding conjecture (conjecture-1) has been proved in . In  authors construct a greedy embedding into the Euclidean plane for all circuit graphs – which is a generalization of -connected planar graphs. Similar result was independently discovered by Angelini, Frati and Grilli .
() Any -connected graph without having a minor admits a greedy embedding into the Euclidean plane.
Also, recently convex greedy embedding conjecture (conjecture-2) has been proved for the case of all planar triangulations  (existentially, using probabilistic methods). Note that the Delaunay triangulation of any set of points in the plane is known to be greedy , and a variant of greedy algorithm (greedy-compass algorithm) of  works for all planar triangulations.
Surely convex greedy embedding conjecture (conjecture-2) implies conjecture-1, however not otherwise. The greedy embedding algorithm presented in [6, 7] does not necessarily produce a convex greedy embedding [11, 12], and in fact the embedding may not even be a planar one. In this work we consider the convex greedy embedding conjecture (conjecture-2).
An alternative way to view the greedy embedding is to consider following path finding algorithm (see Algorithm 1) on a graph and given embedding . The algorithm in every step recursively selects a vertex that is closer to destination than current vertex. To simplify notation we write in place of , when embedding is given.
Clearly, if is a greedy embedding of then for any choice of , we have a distance decreasing path , such that for , . Thus given and , a greedy path finding algorithm succeeds for every pair of vertices in iff is a greedy embedding of .
This simple greedy path finding strategy has many useful applications in practice. Ad hoc networks and sensor nets has no universally known system of addresses like IP addresses. Also, due to resource limitations it is prohibitive to store and maintain large forwarding tables at each node in such networks. To overcome these limitations, geometric routing uses geographic coordinates of the nodes as addresses for routing purposes [13, 14]. Simplest of such strategy can be greedy forwarding strategy as described above (Algorithm-1). However, this simple strategy sometimes fails to deliver a packet because of the phenomenon of “voids” (nodes with no neighbor closer to the destination). In other words the embedding of network graph, provided by the assigned coordinates is not a greedy embedding in such cases. To address these concerns, Rao et al.  proposed a scheme to assign coordinates using a distributed variant of Tutte embedding . On the basis of extensive experimentation they showed that this approach makes greedy routing much more reliable.
Finally, Kleinberg  studied a more general but related question on this direction as: What is the least dimension of a normed vector space where every graph with nodes has a greedy embedding? Kleinberg showed if is a -dimensional normed vector space which admits a greedy embedding of every graph with nodes, then . This implies that for every finite-dimensional normed vector space there exist graphs which have no greedy embedding in . Kleinberg also showed that there exists a finite-dimensional manifold, namely the hyperbolic plane, which admits a greedy embedding of every finite graph.
1.3 Our results
In this work we show that given a -connected planar graph , an embedding of is a planar convex greedy embedding if and only if, in the embedding , weight of the maximum weight spanning tree () and weight of the minimum weight spanning tree () satisfies , for some .
In order to obtain this result we consider a weaker notion of greedy embedding. Weak111Not to be confused with the weaker version of the conjecture. Here weakness is w.r.t. greedy criteria, and not convexity of embedding. greedy embedding allows path finding algorithm to proceed as long as local optima is bounded by a factor. Formally,
Definition 1.2 (Weak greedy embedding).
Let . A –weak greedy embedding of a graph is a planar embedding of with the following property: for every pair of non-adjacent vertices there exists a vertex adjacent to such that .
Surely if admits a -weak greedy embedding then it is greedily embeddable. We show that every -connected planar graph has a -weak greedy convex embedding in with , where is the ratio of maximum and minimum distance between pair of vertices in the embedding of .
Rest of the paper is organized as follows. In Section-2 we present the required definitions which will be used in following sections. In section-3 we define -weak greedy convex embedding and provide a brief outline of the results. Subsequently, in section-4 we derive various results on the -weak greedy convex embedding and show that every -connected planar graph has a -weak greedy convex embedding in with . Finally, in section-5 we derive the new condition on the weight of the minimum weight spanning tree and maximum weight spanning tree that must be satisfied in the greedy convex embedding for every -connected planar graphs. Section-6 contains some concluding remarks.
We will use standard graph theoretic terminology . Let be an undirected graph with vertex set and edge set , where . Given a set of edges , let denote the subgraph of induced by . For a vertex , let denote its neighborhood. A connected acyclic subgraph of is a tree. If , then is a spanning tree. For , –paths and in are internally disjoint if . Let denote the maximum number of pair-wise internally disjoint paths between . A nontrivial graph is -connected if for any two distinct vertices . The connectivity of is the maximum value of for which is -connected.
3 Weak greedy embedding of 3-connected planar graphs
In this section we define -weak greedy convex embedding, and provide an outline of the proof. In rest of the section be a planar convex embedding of which produces a one-to-one mapping from to . We shall specifically consider Tutte embedding ([2, 3, 4, 5]) and a brief description of Tutte embedding has been provided in Appendix-A. Since is fixed, given a graph , we will not differentiate between and its planar convex embedding under viz. .
First let us consider following recursive procedure for –weak greedy path finding given in Algorithm-2.
If is chosen as the minimum value such that at least one branch of this recursive procedure returns then we will call that value of optimal for vertex . Given for a vertex there can be more than one –weak greedy path from to . Let be a subgraph of induced by all vertices and edges of –weak greedy –paths for all possible terminal vertex . Let be any spanning tree of . Surely, has unique –weak greedy –paths for all possible terminal vertex from . We will call optimal weak greedy tree w.r.t vertex . Define . We note that procedure with parameter succeeds to find at least one –weak greedy –paths for all possible vertex pairs . In following our objective will be to obtain a bound on for any -connected planar graph under embedding . To obtain this bound we will use the properties of weak greedy trees.
What follows is a brief description of how we obtain the stated results. In the planar convex embedding of , let weight of an edge be its length i.e. . Define . We obtain a lower and upper bound on the weight of . On the other hand we also obtain a upper bound on the weight of any spanning tree of in its embedding , and a lower bound on the weight of any minimum spanning tree of , . Surely , and from this we derive an upper and a lower bound on . Let be the diameter of , and let minimum edge length in embedding of be . In following (in Section-4.1) we derive that,
Subsequently (in Section-4.2), we show that,
Finally (in Section-4.3), we derive upper and lower bounds on the the weight of as:
Using the fact that , we than show using the bounds described above - that any three connected planar graph has a -weak greedy convex embedding in with , where . Our main result states that given a -connected planar graph , an embedding of is a planar convex greedy embedding if and only if, in the embedding , weight of the maximum weight spanning tree () and weight of the minimum weight spanning tree () satisfies , for some . To establish one side of this implication we use the bounds on the weight of and the upper bound on the weight of the .
4 Bounding the weight of trees
In following we first describe upper bound on the weight of any spanning tree of in its planar convex embedding. In order to obtain this bound we use some ideas from .
4.1 Upper bound on the weight of spanning tree
Given a graph and its planar convex embedding, let be the diameter of and let be any spanning tree of . For let be th edge of (for a fixed indexing of edges). Let be the open disk with center such that is the mid point of , and having diameter . We will call a diametral circle of . Let be the smallest disk (closed) that contains . Define . We have following claim:
is contained into a closed disk having its center coinciding with and having diameter at most .
Let having its center at point . Let be an edge – surely and are points inside . Consider the closed disk centered at the midpoint of having diameter . Let be its center. Since must contain , worst case is when both and are at the boundary of (see Figure-1). Now let be any point on the boundary of . We have:
On the other hand . Hence,
Right side is maximized when , and in that case . ∎∎
Using Lemma-4.1 we can now obtain a bound on . Let denote the circumference of circle , i.e. .
Let be a closed disk in which is contained, where is the smallest disk (closed) that contains . Using Lemma-4.1, and using the fact that is a spanning tree and hence have edges, we have:
4.2 Bound on the weight of minimum weight spanning tree
In the planar convex embedding of let be a minimum weight spanning tree of and let be its weight. In this section we obtain an upper and a lower bound on . Let be the point set given (as images of vertex set) by the embedding. Let be the set of all line-segments corresponding to the all distinct pair of end-points . Also, let be a spanning tree of whose edges are subset of such that weight is minimum ( is a Euclidean minimum spanning tree of the point set ). Surely, : convex embedding produces a straight-line embedding of , and hence the line segments corresponding to the edges of in embedding are also subset of . Let and be vertices having distance . Any would connect and . Hence we have:
In planar convex embedding of ,
We will also require upper bound on the weight of minimum spanning tree for which we have:
In planar convex embedding of ,
Given a graph and its planar convex embedding, let be the diameter of and let be any minimum weight spanning tree of . For let be th edge of (for a fixed indexing of edges). Let be the open disk with center such that is the mid point of , and having diameter . We will call a diametral circle of . Let be the smallest disk (closed) that contains . Define . Recall, using Lemma-4.1 we have that is contained into a closed disk having its center coinciding with and having diameter at most . Let denote the circumference of circle , i.e. , where is a diametral circle of edge . Also, let denote the area of circle , i.e. , where is a circle having diameter . Now,
Now by Lemma-4.1, all the points that we would like to count in are contained in . Except that some of the points that appear on the circumference of more than one circles, must be counted multiple times. In order to bound that we shall use following result from .
Lemma 4.5 (Lemma-2 from ).
For any point , is contained in at most five diametral circles drawn on the edges of the of a point set .
4.3 Bound on the weight of weak greedy trees
Given a graph and its planar convex embedding, let be an optimal weak greedy tree w.r.t a vertex . Let be any leaf vertex of , and consider the –weak greedy –path.
Definition 4.1 (Increasing and decreasing sequence).
Given a graph and its planar convex embedding, for –weak greedy –path , an ordered sequence of vertices of is an increasing sequence of length if holds. Similarly, an ordered sequence of vertices of is a decreasing sequence of length if holds. Usually, we will refer any maximal (by property of monotonically non-decreasing or non-increasing) sequence of vertices as increasing or decreasing sequence.
It is straightforward to observe that if an –path is –weak greedy for , then it has a monotonically non-decreasing sequence of vertices. However, every –path must have a trailing monotonically decreasing sequence that reaches (e.g. see Figure-5(d)). We will call an increasing sequence of a -increasing sequence of length if it is maximal and for holds (with equality for at least one ). We will denote it as , where indicates .
Let be a -increasing sequence of length from a –weak greedy –path such that . Then
Where is the sum of the weight of the edges of .
First let us bound the length of th segment in (see Figure-2). We have , and . Let . We have and . Since
So . Similarly,
Hence, . So starting at a distance from and summing over length sequence, we have for upper bound on :
And for lower bound on we have,
Like , for by we will denote a decreasing sequence of as a -decreasing sequence of length if it is maximal and for , holds (with equality for at least one ), where indicates .
Let be a -decreasing sequence of length such that . Then
A similar calculation as in the proof of Lemma-4.6 shows that the length of th segment is bounded from above by , and from below by . So starting at a distance from and summing over length sequence, we have upper bound on :
And for lower bound on ,
Now, for a path such that is a leaf vertex of the tree , can be written as (where denotes sequential composition), such that , , and for each we have when is odd and when is even. In other words, is a combination of increasing and decreasing sequences with at least one increasing sequence and a trailing decreasing sequence. Also every sequence starts at a distance from , where the immediate previous sequence ends.
Let be a length –weak greedy –path such that is a leaf vertex of the tree . Then
Or the the term of this sum can be written as,
With constraint that , and is odd (since is -weak it can not have only a decreasing sequence, and terminating sequence must be decreasing as is a leaf vertex). For and fixed, second constraint implies that though sum increases if is maximized and is close to , this can not be done without increasing and hence decreasing . So the expression is maximized with and . With this we have from equation-1:
Using equation-2 with , for each , and , we obtain:
Where, the last inequality follows by taking minimum edge length in embedding of as . ∎∎
Finally we bound the weight of -weak greedy spanning tree .
Assume that has many leaf nodes. Then weight of the tree is
Where . In order to obtain the upper bound we observe that is maximized with any one of . Hence using upper bound on from Lemma-4.8 we have: . On the other hand, for the lower bound we have and . Using lower bound on from Lemma-4.8 we have: ∎∎
4.4 Bound on
As stated in the beginning of this section, we now compare the bound on the weight of any spanning tree of with that of as derived in Lemma-4.2, Lemma-4.3 and Lemma-4.9 to obtain an upper and lower bound on .
Let be any three connected planar graph. Then has a -weak greedy convex embedding in with
Also, this bound is achieved by Tutte embedding.
Let be any -weak greedy spanning tree of with respect to vertex . Let be any spanning tree of , and let be any minimum weight spanning tree of . Then using Lemma-4.3, and upper bound on the from Lemma-4.9 we obtain:
Now using we have: