An Improvement to Chvátal and Thomassen's Upper Bound for Oriented Diameter

01/10/2020 ∙ by Jasine Babu, et al. ∙ 0

An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G. An orientation of a graph G is called a strong orientation, if from each vertex there is a directed path to every other vertex. The oriented diameter of a graph G is the smallest diameter among all the strong orientations of G. The maximum oriented diameter of a family of graphs F is the maximum oriented diameter among all the graphs in F. Chvátal and Thomassen gave a lower bound of 1/2d^2+d and an upper bound of 2d^2+2d for the maximum oriented diameter of the family of 2-edge connected graphs of diameter d. We improve this upper bound to 1.373 d^2 + 6.971d-1, which outperforms the former upper bound for all values of d greater than or equal to 8. For the family of 2-edge connected graphs of diameter 3, Kwok, Liu and West obtained improved lower and upper bounds of 9 and 11 respectively. For the family of 2-edge connected graphs of diameter 4, the bounds provided by Chvátal and Thomassen are 12 and 40 and no better bounds were known. By extending the method we used for diameter d graphs, along with a generalized version of a technique used by Chvátal and Thomassen, we have improved this upper bound to 21.

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1 Introduction

An orientation of an undirected graph is an assignment of exactly one direction to each of the edges of . A given undirected graph can be oriented in many different ways (, to be precise, where m is the number of edges). The studies on graph orientations often concern with finding orientations which achieve a predefined objective. Some of the objectives while orienting graphs include minimization of certain distances, ensuring acyclicity, minimizing the maximum in-degree, maximizing connectivity, etc. One of the earliest studies regarding graph orientations were carried out by H.E. Robbins in 1939. He was trying to answer a question posed by Stanislaw Ulam. “When may the arcs of a graph be so oriented that one may pass from any vertex to any other, traversing arcs in the positive sense only?”. This led to a seminal work [1] of Robbins in which he proved the following theorem, “A graph is orientable if and only if it remains connected after the removal of any arc”’.

A directed graph is called strongly connected if it is possible to reach any vertex starting from any other vertex using a directed path. An undirected graph is called strongly orientable if it has a strongly connected orientation. A bridge in a connected graph is an edge whose removal will disconnect the graph. A -edge connected graph is a connected graph which does not contain any bridges. The theorem of Robbins stated earlier says that it is possible for a graph to be strongly oriented if and only if is -edge connected. Though Robbins stated the necessary and sufficient conditions for a graph to have a strong orientation, no comparison between the diameter of a graph and the diameter of an orientation of this graph was given in this study. This was taken up by Chvátal and Thomassen in 1978 [2].

In order to discuss these quantitative results, we introduce some notation. Let be an undirected graph. The distance between two vertices and of , is the number of edges in a shortest path between and . For any two subsets , of , let . The eccentricity of a vertex of is the maximum distance between and any other vertex of . The diameter of is the maximum of the eccentricities of its vertices. The radius of is the minimum of the eccentricities of its vertices. Let be a directed graph and . Then the distance from a vertex to , , is defined as the length of a shortest directed path from to . For any two subsets , of , let . The out-eccentricity of a vertex of is the greatest distance from to a vertex . The in-eccentricity of a vertex of is the greatest distance from a vertex to . The eccentricity of a vertex of is the maximum of its out-eccentricity and in-eccentricity. The diameter of , denoted by , is the maximum of the eccentricities of its vertices. The radius of is the minimum of the eccentricities of its vertices. The oriented diameter of an undirected graph , denoted by , is the smallest diameter among all strong orientations of . That is, G. The oriented radius of an undirected graph is the smallest radius among all strong orientations of . The maximum oriented diameter of the family of graphs is the maximum oriented diameter among all the graphs in . Let denote the maximum oriented diameter of the family of -edge connected diameter graphs. That is, , where is the family of -edge connected graphs with diameter .

The following theorem by Chvátal and Thomassen [2] gives an upper bound for the oriented radius of a graph.

Theorem 1.

[2] Every 2-edge connected graph of radius admits a strong orientation of radius at most .

The above bound was also shown to be tight. In the same paper, they also proved that the problem of deciding whether an undirected graph admits an orientation of diameter is NP-hard. Motivated by the work of Chvátal and Thomassen [2], Chung, Garey and Tarjan [3] proposed a linear-time algorithm to check whether a mixed multigraph has a strong orientation or not. They have also proposed a polynomial time algorithm which provides a strong orientation (if it exists) for a mixed multigraph with oriented radius at most . Studies have also been carried out regarding the oriented diameter of specific subclasses of graphs like AT-free graphs, interval graphs, chordal graphs and planar graphs [4, 5, 6]. Bounds on oriented diameter in terms of other graph parameters like minimum degree and maximum degree are also available in literature [7, 8, 9, 10].

Chvátal and Thomassen [2] has proved that every 2-edge connected graph of diameter admits a strong orientation of diameter at most . They have also proved that every orientation of the Petersen graph has diameter at least . Thus . The following bounds were also given by Chvátal and Thomassen [2].

Theorem 2.

[2] .

By Thoerem 2 we have . In 2010, Kwok, Liu and West [11] improved this bound to . To prove the upper bound of , Kwok, Liu and West partitioned the vertices of into a number of sets based on the distances from the endpoints of an edge which is not part of any -cycle. Our study on the oriented diameter of -edge connected graphs with diameter uses this idea of partitioning the vertex set into a number of sets based on their distances from a specific edge.

Our Results

In this paper we establish two improved upper bounds. Firstly in Section 2, we show that (Theorem 7). This is the first general improvement to Chvátal and Thomassen’s upper bound from 1978. For all , our upper bound outperforms that of Chvátal and Thomassen. Their lower bound still remains unimproved. We do not believe that our upper bound is tight. Secondly in Section 3, for the case of , we further sharpen our analysis and show that (Theorem 13). This is a considerable improvement from , which follows from Chvátal and Thomassen’s general upper bound. Here too, our upper bound is not yet close to the lower bound of given by Chvátal and Thomassen and we believe that there is room for improvement in the upper bound.

2 Oriented Diameter of -edge Connected Diameter Graphs

A dominating set for a graph is a subset of the vertex set of such that every vertex not in is adjacent to at least one vertex of . A subset of the vertex set of is called a -step dominating set of if every vertex not in is at a distance of at most from at least one vertex of . An oriented subgraph of is called a -step dominating oriented subgraph if is a -step dominating set of . To obtain upper bounds for the oriented diameter of a graph with vertices and minimum degree , Bau and Dankelmann [7] and Surmacs [8] first constructed a -step dominating oriented subgraphs of . They used this together with the idea in the proof of Theorem 1 on to obtain the upper bounds of and , respectively, for the oriented diameter of graphs with minimum degree . We are using the algorithm OrientedCore described below to produce a -edge connected oriented subgraph of with some distance guarantees between the vertices in (Lemma 3) and some domination properties (Lemma 5).

2.1 Algorithm OrientedCore

Input:

A -edge connected graph and a specifed edge in .

Output:

A -edge connected oriented subgraph of .

Terminology:

Let be the diameter of , let be the length of a smallest cycle containing in and let . Notice that and . Define . Since is non-empty only if and , we implicitly assume these restrictions on the subscripts of wherever we use it. For a vertex , its level is and its width is . We will always refer to an edge between two different ’s as when either or and (downward or rightward in Figure 1). Moreover the edge is called vertical in the first case and horizontal in the second.

Observations based on the first edge of shortest paths from a vertex to and :

Every vertex , , is incident to a horizontal edge with . Every vertex , , is incident to a horizontal edge with . Every vertex , , is incident either to a horizontal edge with or two vertical edges and with and . Consequently for any in Level , all the shortest path consists of Level horizontal edges only and for any vertex in in Level , all the shortest path consists of Level horizontal edges alone. For any vertex in Level , all the shortest path consists of horizontal edges in levels and and exactly one vertical edge; while all the shortest path consists of horizontal edges in levels and and exactly one vertical edge.

Stage .

Initialise to be empty. For each vertical edge with and , and for each shortest path and shortest path , do the following: Let be the path formed by joining , the edge and . Orient the path as a directed path from to and add it to . Notice that even though two such paths can share edges, there is no conflict in the above orientation since, in Stage , every vertical edge is oriented downward, every horizontal edge in Level  is oriented rightward and every horizontal edge in levels and is oriented leftward.

Stage .

For each vertical edge with and not already oriented in Stage , and for each shortest path and shortest path do the following: Let be the last vertex in (nearest to ) that is already in and let be the subpath of from to . Similarly let be the first vertex in (nearest to ) that is already in and let be the subpath of from to . Let be the path formed by joining , the edge and . Orient the path as a directed path from to and add it to . Notice that does not share any edge with a path added to in Stage , but it can share edges with paths added in earlier steps of Stage . However there is no conflict in the orientation since, in Stage , every vertical edge is oriented downward, every horizontal edge in Level  is oriented rightward, every horizontal edge in Level  is oriented leftward, and no horizontal edges in Level  is added.

Stage .

Finally orient the edge from to and add it to . This completes the construction of , the output of the algorithm.

Distances in

First we analyse the (directed) distance from and to of vertices added to in Stage . The following bounds on distances in follow from the construction of each path in Stage 1. Let be any vertex that is added to in Stage . Then

(1)
(2)

It is easy to verify the above equations using the facts that is part of a directed path of length at most (at most if ) in .

No new vertices from Level  or are added to in Stage . Still the distance bounds for vertices added in Stage  are slightly more complicated since a path added in this stage will start from a vertex in Level  and end in a vertex in Level , which are added to in Stage . But we can complete the analysis since we already know that and where is such that from the analysis of Stage 1. Let be any vertex that is added to in Stage . Then

(3)

The distance from to in is not affected even though we trim the path at since already has a directed shortest path to from Stage . Hence

(4)

The first part of the next lemma follows from taking the worst case among (1) and (3). Notice that and when . New vertices are added to in Stage 2 only if . The second part follows from (2) and (4). The subsequent two claims are easy observations.

Lemma 3.

Let be a -edge connected graph, be any edge of and let be the oriented subgraph of returned by the algorithm OrientedCore. Then for every vertex we have

(5)
(6)

Moreover, and .

We can see that if is non-empty, then all the vertices in are captured into .

Notice that when , and are non empty. Thus the bound on the diameter of follows by the triangle inequality and the fact that the worst bounds for and from Lemma 3 are when and . Hence we have the following corollary.

Corollary 4.

Let be a -edge connected graph, be any edge of and let be the oriented subgraph of returned by the algorithm OrientedCore. If the length of the smallest cycle containing is greater than or equal to 4, then the diameter of is at most .

Domination by

Let us call the vertices in as captured and those in as uncaptured. For each let and denote the captured and uncaptured vertices in level i, respectively. Since contains every level vertex incident with a vertical edge, separates from rest of . Let denote the maximum distance between a vertex in and the set . Let and for any distinct . The distance is bounded above by , the diameter of , and bounded below by . Hence for every distinct .

For any vertex , the last Level vertex in a shortest (undirected) path is in . Hence if Level is non-empty then . In order to bound and , we take a close look at a shortest cycle containing the edge . Let with and . Each is in when , if and when . Let . The Level vertex is special since it is at a distance from Level and thus . For every vertex , the distance is bounded above by and below by . Hence . Similarly we can see that .

Putting all these distance bounds on domination together, we get the next lemma.

Lemma 5.

Let be a -edge connected graph, be any edge of and let be the oriented subgraph of returned by the algorithm OrientedCore. For each , let denote the maximum distance of a level vertex not in to the set of level vertices in . Then , and for any distinct , .

2.2 The Upper Bound

Consider a -edge connected graph with diameter . Let denote the smallest integer such that every edge of a graph belongs to a cycle of length at most . Huang, Li, Li and Sun [12] proved the following theorem.

Theorem 6.

[12]

where is the radius of and .

We know that and hence we have as our first bound. Let be an edge in such that the length of a smallest cycle containing is . If , then . So we assume . By Corollary 4, has an oriented subgraph with diameter at most . Moreover by Lemma 5, is a -step dominating subgraph of . Let be a graph obtained by contracting the vertices in into a single vertex . We can see that has radius at most . Thus by Theorem 1, has a strong orientation with radius at most . Since , we have . Notice that and do not have any common edges. Hence has an orientation with diameter at most by combining the orientations in and . Let . Hence we get . We can see that the dominant term in the first bound is while the dominant term in the second bound is at most . Notice that . Thus by optimizing for in the range we obtain the following theorem.

Theorem 7.

.

For any , the above upper bound is an improvement over the upper bound of provided by Chvátal and Thomassen.

3 Oriented Diameter of -edge Connected Diameter Graphs

In this section is an arbitrary -edge connected diameter graph. We will show that the oriented diameter of is at most and hence . The following lemma by Chvátal and Thomassen [2] is used when .

Lemma 8.

[2] Let be a 2-edge connected graph. If every edge of lies in a cycle of length at most , then it has an orientation such that

Hence if all edges of the graph lie in a -cycle or a -cycle, the oriented diameter of will be at most . Hence we can assume the existence of an edge which is not part of any -cycle or -cycle as long as we are trying to prove an upper bound of or more for . We apply algorithm OrientdCore on with the edge to obtain an oriented subgraph of . Figure 1 shows a course representation of .

Figure 1: A course representation of which shows the orientation of edges between various subsets of . A single arrow from one part to another indicates that all the edges between these parts are oriented from the former to latter. A double arrow between two parts indicates that the edges between the two parts are oriented in either direction or unoriented. An unoriented edge between two parts indicate that no edge between these two parts are oriented.

3.1 Oriented Diameter and -step Domination Property of

Since is the oriented subgraph of returned by the algorithm OrientedCore we can apply Corollary 4. Since the smallest cycle containing is of length greater than or equal to we can see that the diameter of is at most . Moreover from equations 5 and 6 of Lemma 3 we get the upper bounds on the distances of in Table 1. Hence we have the following corollary.

Corollary 9.

. Moreover , and obey the bounds in Table 1.


for in
1 2 3 2 5 6 9 8 7
7 6 5 2 5 4 1 2 3

Table 1: Upper bounds on the distances of
Remark 1.

If , then is empty. Moreover if is non-empty, then all the vertices in are captured into .

Furthermore, applying Lemma 5 on shows that is a -step dominating subgraph of . Let be a graph obtained by contracting the vertices in into a single vertex . We can see that has radius at most . Thus by Theorem 1, has a strong orientation with radius at most . Since , we have . Since and do not have any common edges we can see that has an orientation with diameter at most by combining the orientations in and . But we further improve this bound to by constructing a -step dominating oriented subgraph of . We propose the following asymmetric variant of a lemma by Chvátal and Thomassen for the construction and anlysis of .

3.2 Asymmetric Chvátal-Thomassen Lemma

For any subset of , let denote the set of all vertices with an edge incident on some vertex in . Let be a subgraph of . An ear of in is a sequence of edges such that and none of the vertices and none of the edges in this sequence are in . In particular we allow .

Lemma 10 (Asymmetric Chvátal-Thomassen Lemma).

Let be an undirected graph and let such that

  1. is a -step dominating set in ,

  2. is -edge connected, and

  3. is a -step dominating set of .

Then there exists an oriented subgraph of such that

  1. and hence is a -step dominating set of , and

  2. , we have and either or .

Proof.

We construct a sequence of oriented subgraphs of as follows. We start with and add an oriented ear in each step. Let . If , then we stop the construction and set . Since is a -step dominating set of , the first conclusion of the lemma is satisfied when the construction ends with . Otherwise, let and let be a neighbour of in . Since is -edge connected, there exists a path in from to . Let be a shortest path in with the additional property that once hits a vertex in an oriented ear that was added in a previous step, continues further to along the shorter arm of . It can be verified that is still a shortest path in . The ear is the union of the edge and the path . If hits without hitting any previous ear, then we orient as a directed path from to . If , then we orient as a directed path by extending the orientation of . Notice that, in both these cases, the source vertex of is in . We add to to obtain .

Let with and be the ear added in the -th stage above. Since is a shortest path in and since is a -step dominating set, . Moreover, if , then since is a -step dominating set. These bounds on the length of along with the observation that the source vertex of is in , verifies the second conclusion of the lemma. ∎

Remark 2.

If we flip the orientation of we get the bounds and either or , in place of Conclusion (ii) of Lemma 10.

Setting in Lemma 10 gives the original Chvátal-Thomassen Lemma. Notice from the above proof that, in this case .

Lemma 11 (Chvátal-Thomassen Lemma).

Let be an undirected graph and let such that

  1. is a -step dominating set in , and

  2. is -edge connected.

Then there exists an oriented subgraph of such that

  1. is a -step dominating set of , and

  2. , we have and .

Let be any -edge connected graph with radius . Chvátal and Thomassen showed that by applications of Lemma 11; starting with , where is any central vertex of and in each subsequent application being the vertex-set of the oriented subgraph returned by the current application.

3.3 A Dominating Oriented Subgraph of

Let be the oriented subgraph of returned by the algorithm OrientedCore. We will add further oriented ears to to obtain a -step dominating oriented subgraph of . We have already seen that is a -step dominating oriented subgraph of . By Lemma 5, we also have for any distinct .

Now consider the first case where we have vertices in Level which are at a distance from . Notice that in this case and hence . Let , and . Notice that is a cut-set separating from the rest of and hence the graph is -edge connected. Since , we can see that is -step dominating in . Since and we see that is a 1-step dominating subgraph in and therefore we can apply Lemma 10 on . Every edge of the oriented subgraph of obtained by applying Lemma 10 is reversed to obtain the subgraph . Now consider the vertices captured into . From Lemma 10 and Remark 2, we get the following bounds of and either or , . Here and from Table 1, we have the bounds , and . Hence , . Since and we also have , . Let and let now denote . By the above discussion, we get the bounds in Table 2 for and when . Moreover, .


for in