Tree Spanners of Simple Graphs

Transcription

1 Tree Spanners of Simple Graphs by Ioannis E. Papoutsakis A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Computer Science University of Toronto c Copyright by Ioannis Papoutsakis 2013

2 Tree Spanners of Simple Graphs Doctor of Philosophy, 2013 Ioannis E. Papoutsakis Graduate Department of Computer Science University of Toronto Abstract A tree t-spanner T of a simple graph G is a spanning tree of G, such that for every pair of vertices of G their distance in T is at most t times their distance in G, where t is called a stretch factor of T in G. It has been shown that there is a linear time algorithm to find a tree 2-spanner in a graph; it has also been proved that, for each t > 3, determining whether a graph admits a tree t-spanner is an NP-complete problem. This thesis studies tree t-spanners from both theoretical and algorithmic perspectives. In particular, it is proved that a nontree graph admits a unique tree t-spanner for at most one value of stretch factor t. As a corollary, a nontree bipartite graph cannot admit a unique tree t-spanner for any t. But, for each t, there are infinitely many nontree graphs that admit exactly one tree t-spanner. Furthermore, for each t, let U(t) be the set of graphs being the union of two tree t-spanners of a graph. Although graphs in U(2) do not have cycles of length greater than 4, graphs in U(3) may contain cycles of arbitrary length. It turns out that any even cycle is an induced subgraph of a graph in U(3), while no graph in U(3) contains an induced odd cycle other than a triangle; graphs in U(3) are shown to be perfect. Also, properties of induced even cycles of graphs in U(3) are presented. For each t > 3, though, graphs in U(t) may contain induced odd cycles of any length. Moreover, there is an efficient algorithm to recognize graphs that admit a tree 3-spanner of diameter at most 4, while it is proved that, for each t > 3, determining whether a graph admits a tree t-spanner of diameter at most t+1 is an NP-complete problem. It is not known if it is hard to recognize graphs that admit a tree 3-spanner of general diameter; however integer programming is employed to provide certificates of tree 3-spanner inadmissibility for a family of graphs. ii

3 Acknowledgments During my graduate studies at the University of Toronto, I have had many interesting and enlightening meetings with my supervisor Derek Corneil. I am grateful to him for his teaching and support. I am also grateful to Michael Molloy for his instructive remarks and lectures since the beginning of my graduate studies. Rudi Mathon, Michael Carter and Udi Rotics, together with Derek Corneil and Michael Molloy, have been members of my advisory committee. I thank each and all of them for their comments and advice on my papers and presentations. I also thank Eric Mendelsohn for participating in the final thesis examination committee. Special thanks go to Andreas Brandstädt from the University of Rostock for reviewing my thesis and for his advice on research directions during his visit for the final thesis examination. Bruce Reed, Ekkehard Köhler, Feodor Dragan and Frank Van Bussel read preliminary drafts of parts of this document and made helpful comments. For formatting this document, I have used parts of L A TEX files provided by Francois Pitt. I also thank people at the University of Toronto and especially at the Department of Computer Science for their assistance and cooperation. For my graduate studies, I have received financial support from the University of Toronto and particularly from the Connaught Fund. iii

4 Contents 1 Introduction Tree spanners starting from scratch A review of the literature on spanners An application in some detail Definitions Graph theory Algorithms Some tools Overview Unique tree spanners Tree t-spanner vs t-star Stretch factor and uniqueness A sufficient condition for uniqueness Examples Stretch factor is an even number Stretch factor is an odd number Unique tree t-spanners of big diameter Pairs of tree spanners How many tree t-spanner edges can a cycle contain? The union of any two tree 2-spanners Induced cycles in the union of any two tree 3-spanners iv

5 The structure of induced cycles The union of two tree 3-spanners is a perfect graph Pairs of tree t-spanners for t Small diameter tree spanners Shortest paths spanning trees Tree 3-spanners of diameter Tree t-spanners of diameter t+1 for t Stretch factor equals The remaining values The situation for each value of the stretch factor Graphs that do not admit a tree 3-spanner Sets of valid inequalities for each stretch factor Properties of tree 3-spanner admissibility Necessary edges Induced cycle constraints Tree 3-spanners that contain a given subtree Certificates for tree 3-spanner inadmissibility The t = 2 case Some guidelines for future research Is it hard to find a tree 3-spanner of a graph? Properties of solutions to the tree 3-spanner problem Some open problems Bibliography 86 Appendix 90 v

6 Chapter 1 Introduction 1.1 Tree spanners starting from scratch A simple graph G consists of a set of vertices V(G) and a set of edges E(G) joining unordered pairs of distinct vertices in V(G). An edge joining vertices u and v is written as uv, where u and v are the endpoints of the edge. A graph is empty if and only if V(G) =. Unless otherwise stated, a graph is a simple graph. A graph gives rise to a distance function between its vertices. A subgraph of G is a graph which consists of vertices and edges of G only; a spanning subgraph of G is a subgraph of G that contains all the vertices of G. In this thesis, we study relations between the distance function of a graph G and the distance function of special spanning subgraphs of G, which are called spanners. Paths are used to define the distance function of a graph and we consider spanners that are trees. A path P of length k is a nonempty graph with a vertex set V(P) = {v 0,...,v k } and a possibly empty edge set E(G) = {e 1,...,e k }, such that e i = v i 1 v i, for all i from 1 to k. Vertices v 0 and v k are the only vertices of P that are not incident to two edges of P and they are the endpoints of P; a path of length 0 has only one endpoint. The length of a path P is denoted as P. Given a graph G, a subgraph of G which is a path with endpoints u and v is simply called a u, v-path in G. Two (possibly identical) vertices u and v of a graph G are at distance d from each other in G if and only if there is a u, v-path of length d in G and there is not any u, v-path 1

7 in G of length strictly less than d. Thus, if there is a u, v-path in a graph G, then there is a unique number d, such that u and v are at distance d from each other in G; we denote the distance in G between u and v as d G (u,v). A tree T is a nonempty graph, such that for every pair of vertices u, v of T there is a unique u, v-path in T. Definition A graph T is a tree t-spanner of a graph G if and only if T is a subgraph of G that is a tree and, for every pair u and v of vertices of G, if u and v are at distance d from each other in G, then u and v are at distance at most t d from each other in T. If T is a tree t-spanner of G, then all the vertices of G are also vertices of T, because each vertex of G is at distance 0 from itself in G, i.e. T is a spanning tree of G. We refer to t as a stretch factor of tree t-spanner T of G. Obviously, if T has stretch factor t in G, then T also has stretch factor t+1 in G. The only graph that admits a tree 0-spanner is the one vertex graph. Also, a graph admits a tree 1-spanner if and only if it is a tree. Generally, given a graph and a number t, there is a set of trees that are the tree t-spanners of the graph. So, for every number t there is a mapping F t from the set of graphs to the set of trees, such that T belongs to F t (G) if and only if T is a tree t-spanner of G. Since for each t a tree t-spanner of a graph is also a tree (t + 1)-spanner of this graph, we know that F t (G) F t+1 (G) for every graph G. Note that if g is an isomorphism 1 between graphs G 1 and G 2, then g sends each tree t-spanner of G 1 to a tree t-spanner of G 2. To illustrate the concept of F t (G) consider the following proposition, which is also mentioned in [22]. Proposition If G is a bipartite graph, then F 2t 1 (G) = F 2t (G) for every positive t. Proof. Let t be any positive number and T any tree (2t)-spanner of a bipartite graph G. Assume that T is not a tree (2t 1)-spanner of G. Then, there is an edge uv in G such that d T (u,v) = 2t. So, edge uv together with the path of T from u to v forms a 1 Terms written in italics are either defined immediately or appear in the Appendix. 2

8 cycle of odd length in G, which is a contradiction, since G is bipartite. Hence, T is a (2t 1)-spanner of G and, therefore, F 2t 1 (G) = F 2t (G). As a corollary, a nontree bipartite graph G does not admit tree 2-spanners, since F 1 (G) = F 2 (G) and trees are the only graphs that admit a tree 1-spanner. 1.2 A review of the literature on spanners There are applications of spanners in a variety of areas, such as distributed computing [3, 34], communication networks [31, 32], motion planning and robotics [2, 12] and phylogenetic analysis [4]. Furthermore, spanners are used in embedding finite metric spaces in graphs approximately [35]. Tree spanners For each t, the tree t-spanner problem is to determine whether a given graph admits a tree t-spanner. The complexity status of these problems changes with the stretch factor t. Note that, for each t, the tree t-spanner problem belongs to NP. When t = 1, the problem is easy, since all the edges of the input graph have to be in a tree 1-spanner of the graph. So, a graph G admits a tree 1-spanner if and only if G is a tree. When t = 2, the problem becomes interesting. There is a linear time algorithm to check ifagraphadmitsatree2-spanner [6,10,8]. Notethattree2-spanner admissible graphs coincide with trigraphs introduced by Bondy [6] in his work on cycle double covers. A cycle double cover of a graph G is a collection C of cycles of G, such that each edge of G belongs to exactly two cycles of C. Note that it is conjectured that every 2-edge connected graph admits a cycle double cover. In [6, 24], it is proved that every 2-edge connected trigraph on n vertices admits a cycle double cover C, such that C = n 1. Additionally, in [6], it is conjectured that each of the remaining 2-edge connected graphs (i.e. graphs that admit a tree t-spanner for t 3 but do no admit a tree 2-spanner) admits a cycle double cover which consists of strictly less than n 1 cycles. 3

9 Finally, the complexity status of the tree 3-spanner problem is unresolved, while, for each t 4, the tree t-spanner problem is NP-complete [10, 8]. Tree t-spanners (t 3) have been studied for various families of graphs. If a connected graphis a cographor a splitgraphor thecomplement ofabipartite graph, then it admits a tree 3-spanner [8]. Also, all convex bipartite graphs have a tree 3-spanner, which can be constructed in linear time [37]. Efficient algorithms to recognize graphs that admit a tree 3-spanner have been developed for interval, permutation and regular bipartite graphs [27], planar graphs [16] and directed path graphs [23]. Moreover, every strongly chordal graph admits a tree 4-spanner, which can be constructed in linear time [7]; note that, for each t, there is a connected chordal graph that does not admit any tree t-spanner. In [16], it is also shown that it is NP-hard to determine the minimum t for which a planar graph admits a tree t-spanner. General spanners The notion of stretch factor can be generalized to nontree subgraphs of a graph. A subgraph H of a graph G has stretch factor t in G if and only if d H (u,v) t d G (u,v), for every pair of vertices u and v in G. In this manner, H is called a t-spanner of G. Since for t > 0 any graph is a t-spanner of itself, we look for t-spanners of a given graph which have as few edges as possible. If T is a tree t-spanner of a graph G, then any t-spanner of G which is not a tree has strictly more edges than T; but, given a t, there are graphs that do not admit a tree t-spanner. The problem of determining, for a given graph G and two positive integers t and m, whether G has a t-spanner with m or fewer edges is an NP-complete problem [33]. In [25, 9, 37, 20, 15, 21, 22], there are results towards this direction. Moreover, one can assign weights to the edges of a graph. For a weighted graph G, the length of a path in G is equal to the sum of the weights of the edges of the path. So, in this case, vertices u and v are at distance d G (u,v) in graph G if and only if there is a u, v-path of length d G (u,v) in G and there is no u, v-path in G of smaller length. Again, a subgraph H of a weighted graph G has stretch factor t if and only if d H (u,v) t d G (u,v) for every pair of vertices u and v in G; H is called a t- 4

10 spanner of G. For each rational t > 1, the problem of determining if a weighted graph admits a spanning tree with stretch factor t is an NP-complete problem [8, 10]. Also, t-spanners that are not necessarily trees are studied in [11, 1] for weighted graphs. In general, a subgraph H of a graph G is an f(x)-spanner of G if and only if d H (u,v) f(d G (u,v)) for every pair of vertices u and v in G, where f is a function on one variable. Cases where f is a linear function are studied in [26, 7]. Another possible generalization is to allow spanners of a graph to include edges which are not in the graph [14, 7]. For example, for every connected chordal graph G, there is a spanning tree T of the square of G, such that d T (u,v) d G (u,v)+2, for every pair of vertices u and v in G [7]. An application in some detail This application arises from the area of distributed computing, where many interconnected processors perform a computational task. A network of processors consists of a set of processors and a set of communication channels between pairs of these processors. Naturally, the communication graph G of such a network is a graph with vertices the processors of the network, such that edge uv is in G if and only if there is a channel in the network between processors u and v. A network of processors is used to execute distributed algorithms. An algorithm which is designed based on the restriction that processors communicate over the channels of the network only at specific time slots is called a synchronous algorithm. On one hand, this restriction simplifies the design of a distributed algorithm, since there is a global handling of the dependences among the computations of the processors. On the other hand, such a restriction may prevent the full use of the resources of the network, since some processors may be idle until the next time slot comes, during which processors can transmit their output and receive new input. An algorithm for which this restriction does not apply is called asynchronous. In practice, an asynchronous algorithm is usually faster than an equivalent synchronous one, but the design and analysis of an asynchronous algorithm is usually more complicated. The main idea in [3] is to design a procedure, called a synchronizer, which runs 5

11 locally in each processor of a network and can be added as a subroutine to any synchronous algorithm designed for this network. Now, a particular execution of the resulting code behaves as an asynchronous algorithm since processors communicate to each other not only on specific time slots. Therefore, one can enjoy both the simplicity of designing a synchronous algorithm and the efficiency of an asynchronous one, by paying the extra cost of the synchronizer. The design of a synchronizer does not depend on the various algorithms it may assist, but does depend upon the structure of the network of processors. Let G be the communication graph of a network. Consider a t 0 -spanner H of G, such that H has m 0 edges, there is not any t 0 -spanner in G with less than m 0 edges and t 0 is the minimum t that G admits a t-spanner with at most m 0 edges. Then, H gives rise to a nearly optimal synchronizer for this network [34]. Therefore, it is essential to study t-spanners in communication graphs of widely used networks of processors [34, 25, 15, 21]. For example, hypercubes are such graphs, where hypercube Q on n vertices admits a 3-spanner with less than 4n edges and does not admit any 3-spanner with less than 3n 1 edges [34, 15]; note that a 2-spanner of Q has to 2 contain all the edges of Q. Also, Q does admit a tree (2log 2 n 1)-spanner (with n 1 edges), but it does not admit any tree spanner of smaller stretch factor, where we have translated results of [19, 18] (see also [38]) into spanner terminology. 1.3 Definitions Definitions which have been invented for the needs of this thesis will be presented in the next section or in the body of the thesis. Also, the appendix contains some additional definitions. Graph theory Most of our terminology follows [38]. In the first section of the thesis, we have already defined basic terms of graph theory necessary to introduce the notion of a tree t-spanner of a graph. Frequently, we refer to distances between subgraphs of a 6

12 graph and here we define this generalization. For any two subgraphs A and B of a graph G, a path from A to B in G is any u, v-path P of G, such that A, P share only vertex u and B, P share only vertex v. Now, two subgraphs A and B of a graph G are at distance d from each other in G if and only if there is a path in G from A to B of length d and there is not any shorter path between them in G. Moreover, the diameter of a graph G is max u,v V(G) d(u,v). A vertex u in a graph G is usually adjacent to some other vertices of G, which are the neighbors of u in G. For every vertex u in a graph G, the set of vertices x of G, such that edge xu is in G, forms the neighborhood N G (u) of u in G; moreover, N G [u] = N G (u) {u}istheclosed neighborhoodofuing. Thenotionofneighborhood can be generalized to subsets of the vertex set of a graph: if A is a subgraph of a graph G, then N G [A] = u V(A) N G [u] and N G (A) = N G [A]\V(A). The degree d G (u) of a vertex u in a graph G is equal to N G (u). A graph is regular if and only if all its vertices have the same degree. Also, (G) is the maximum degree in a graph G. Some graphs are distinguished because of their frequent use and their appearance. A complete graph or clique is a graph in which any two vertices are joined by an edge; there is only one clique on n vertices (up to isomorphisms), which is denoted as K n. Also, a cycle consists of a path of length at least 2 and edge uv, where u and v are the endpoints of the path; the number of edges in a cycle C is the length of the cycle, which is denoted as C. Again, there is only one cycle on n vertices, which is denoted as C n ; note that C 3 is also called a triangle. In case a graph G contains a cycle, the girth of G is the length of a shortest cycle in G. An acyclic graph, i.e. a graph without any cycles, is called a forest. An independent or stable set S in a graph G is a set of vertices of G, such that there is not any edge of G between vertices of S; for example, an even cycle C 2n has exactly two maximum independent sets, each of size n. A graph is bipartite if and only if its vertex set can be partitioned into two independent sets. Note that a graph G is bipartite if and only if G does not contain any cycle of odd length as a subgraph [38]. A graph G is connected if and only if for every pair of vertices u and v of G there is a u, v-path in G; naturally, a graph which is not connected is called disconnected. 7

13 Moreover, a 2-connected graph G is a connected graph with at least 3 vertices, such that G\{u} remains a connected graph, for every vertex u of G; similarly, a 2-edge connected graph G is a connected graph with at least one edge, such that G \ {e} remains a connected graph, for every edge e of G. A vertex in a graph G whose removal from G results in a disconnected graph is a cut-vertex of G. Generally, a subgraph S of a graph G is a separating set or a vertex cutset of G if and only if G\V(S) is a disconnected graph. There is a variety of subgraphs in a graph G. An induced subgraph H of a graph G is a subgraph of G, such that every edge of G joining vertices of H is also an edge of H; the subgraph of G induced by the vertices in H is denoted as G[H]. A maximal connected subgraph of a graph G is called a component of G; for example, a disconnected graph has at least two components. Also, a block of a graph G is a maximal connected subgraph of G that has no cut-vertex. Some properties of blocks are the following. Any two blocks of a graph share at most one vertex; also, T is a spanning tree of a connected graph G if and only if T is the union of one spanning tree from each block of G. Coloring a graph is a major theme in graph theory. A proper coloring of a graph G is an assignment of colors to the vertices of G, such that any two adjacent vertices of G have different colors, where the minimum number of colors needed to properly color G is the chromatic number χ(g) of G. Obviously, the chromatic number of a clique K n is equal to n, since any two vertices of K n must be assigned different colors in every proper coloring of K n. Moreover, if a graph G contains a clique on k vertices as a subgraph, then χ(g) k; the maximum k for which a graph G contains a K k is called the clique number ω(g) of G. The notion of perfect graphs is based on the interaction between cliques in a graph and the chromatic number of subgraphs of the graph. A graph G is perfect if and only if χ(h) = ω(h) for every induced subgraph H of G. Towards recognizing the family of perfect graphs, it is conjectured that if a graph has no induced subgraph isomorphic to either cycle C p whose length p is odd and at least five or the complement C p of such a cycle, then the graph is perfect [5]. This is widely known as the Strong Perfect Graph Conjecture; graphs that satisfy the 8

14 requirements of the conjecture are called Berge graphs. Note that the complement G of any graph G is graph K \E(G), where K is the complete graph with vertex set V(G). Algorithms Here, we intend to provide simplified versions of concepts in computational complexity that we use in the thesis. In [29, 17], there is a detailed exposition of these concepts. We may assume that all possible instances to our problems form a set Σ. Then, each decision problem defines a language being the instances for which the answer to the problem is positive. For example, the language of the tree 3-spanner problem is the set of tree 3-spanner admissible graphs; often, we do not distinguish the notion of a problem from the notion of a language. Now, a decision problem can be solved efficiently if and only if there is a deterministic Turing Machine which decides the language corresponding to the problem in polynomial time. For our purposes, a sketchy description of a deterministic Turing Machine that decides the language of a problem in polynomial time suffices to prove that the problem can be solved efficiently. There are problems for which it is not known if they can be solved efficiently and for such problems we consider another type of Turing Machine. A problem belongs to the complexity class NP if and only if there is a nondeterministic Turing Machine which decides the language of the problem in polynomial time. One way to prove that a language belongs to NP is the following. Language L belongs to NP if and only if there is a deterministic Turing Machine M, which receives an ordered pair of inputs x and y, runs in time polynomial in the size of x and accepts or rejects, such that L = {a Σ : there exists b Σ such that M on input a and b accepts}. For example, consider an algorithm which on input G and T accepts in time polynomial in the size of G if and only if T is a tree 3-spanner of G. Then, the set of tree 3-spanner admissible graphs belongs to NP, since precisely tree 3-spanner admissible graphs admit a tree 3-spanner. Here, a tree 3-spanner of a graph is a certificate that the graph is tree 3-spanner admissible. An approach to study problems in NP is to determine relations between them 9

15 using deterministic Turing Machines. A language B is polynomially reducible to language A if and only if there is a function f from Σ to itself, such that x B if and only if f(x) A, for every x in Σ, where function f can be efficiently computed, i.e. there is a deterministic Turing Machine which given any x in Σ outputs f(x) in time polynomial in the size of x. A language A is hard (or NP-hard) if and only if every problem in NP is polynomially reducible to A. Moreover, a problem in NP which is also hard is an NP-complete problem. Now, Cook s theorem establishes the existence of NP-complete problems. Note that in order to prove that a problem A is hard it suffices to polynomially reduce an NP-complete problem to A. A mirroring of class NP is class co-np, where a language L belongs to co-np if and only if language Σ \ L belongs to NP. For example, we know that the tree 3-spanner problem belongs to NP, so the set of graphs that do not admit a tree 3- spanner is a language in co-np. It is conjectured that NP-hard problems are not in class co-np and, therefore, if a problem belongs to NP co-np, then it is expected that this problem is not hard. 1.4 Some tools There are various characterizations of trees, which we present here. Theorem ([38]) For a simple graph G with n 1 vertices, the following are equivalent: 1. G has exactly one u, v-path, for every pair of vertices u, v of G, i.e. G is a tree. 2. G is connected and has no cycles. 3. G is connected and has n 1 edges. 4. G has n 1 edges and no cycles. The following observation simplifies the definition of a tree t-spanner and it is actually used as the definition of a tree t-spanner in [34]: 10

16 Proposition ([10, 8, 33]) A spanning tree T of a graph G is a tree t-spanner of G if and only if d T (u,v) t, for every edge uv of G. Using this proposition, we can prove that for any graph G, T is a tree x-spanner of G if and only if T is a tree x -spanner of G [10, 8], where x is any nonnegative rational number; observe that all distances in G are integral. Thus, it suffices to examine integral values of the stretch factor 2. There is an extensive study of the tree 2-spanners of a graph in [6, 10, 8] and we discuss some of these results in this paragraph. Let G be a tree 2-spanner admissible graph. If an edge of G appears in all the tree 2-spanners of G, then it is a forced edge. Assume that G does not have any cut-vertex. On one hand, if G does not have any forced edges, then every tree 2-spanner of G is a star, i.e. a vertex adjacent to all the remaining vertices of G with edges of the tree 2-spanner. On the other hand, if G has some forced edges, then the forced edges of G form a tree, which is called the skeleton tree S(G) of G. A component of G \ S(G) is called a compound leaf. Also, all the vertices of a compound leaf are adjacent in G to both endpoints of an edge of S(G) and they are not adjacent to any other vertex of S(G). The set of edges between all the vertices of a compound leaf and one of the endpoints of the edge corresponding to this compound leaf is called a leafstalk. So, each compound leaf has exactly two leafstalks. The following theorem shows the importance of the skeleton tree. Theorem ([10, 8]) Let G be a tree 2-spanner admissible graph that has at least one forced edge and no cut-vertex. A spanning tree T is a tree 2-spanner of G if and only if it is obtained from the skeleton tree S(G) by adding exactly one leafstalk for each compound leaf of G. Now, consider arbitrary values of the stretch factor. A path of even length has a central vertex, while a path of odd length has a central edge. We take into account this parity fact to define a class of trivially tree t-spanner admissible graphs. 2 Of course, in case that we deal with weighted graphs then this argument does not work, since the path of a tree x-spanner of a weighted graph G between a pair of adjacent in G vertices may be longer by x x than the path of a tree x -spanner of G between these two vertices. For example, a triangle with weights 2,2,1 on its edges is not tree 3 2 -spanner admissible, but it is tree 3 2-spanner admissible. Additionally, the existence of such examples is guaranteed, since the tree t-spanner problem for weighted graphs is NP-complete for any rational t > 1 [10, 8]. 11

17 Definition A t-center K of a graph G is a subgraph of G consisting exactly of either a vertex when t is even, or a pair of adjacent in K vertices when t is odd, such that for all u in G, d G (K,u) t. 2 Clearly, for any t-center K, we see that K = E(K) = t mod 2. Assume that a graph G contains a t-center K. Any Breadth-First Search tree T of G starting from K has the property that d T (K,u) = d G (K,u), for every vertex u of G [38]. Therefore, since K is a t-center of G, T is a tree t-spanner of G; observe that the distance in T between any pair of vertices u and v is at most equal to the distance from u to K plus K plus the distance from K to v. Graphs that admit a t-center are defined to be t-stars: Definition A graph G is a t-star if and only if G is the one vertex graph or G admits a t-center. When we write just star, we mean a 2-star. Note that the one vertex graph is a (2t+1)-star for any t, but it does not admit a (2t+1)-center. Also, observe that the only 0-star is the one vertex graph and the only 1-stars are the one vertex graph and the one edge graph. If k t, then a k-star is also a t-star. Moreover, a t-star is a connected graph. Also, every connected graph is a t-star, for some t. For example, a path of length t is a t-star and has a unique t-center. If a graph is a t-star, then at least one spanning tree of the graph has diameter at most t: Lemma A graph G is a t-star if and only if G admits a spanning tree of diameter at most t. Proof. On one hand, assume that G is a t-star. If G is the one vertex graph, then the graph itself is such a spanning tree. Otherwise, let K be a t-center of G. Then, as we mentioned earlier, any Breadth-First Search tree of G starting from K has diameter at most t. On the other hand, assume that G is not a t-star for some t. We prove that the diameter of any spanning tree of G is strictly greater than t. Towards a contradiction, assume that G admits a spanning tree T of diameter d such that d t. Consider a 12

18 longest path P of T, then d = P. Also, P itself is a d-star with a unique d-center K. Let P 1 and P 2 be the subpaths of P from K to u and from K to v, respectively, where u and v are the endpoints of P. Then, P 1 = P 2 = d 2. Let x be an arbitrary vertex of G. Since T is a spanning tree of G, there is a path P from x to K in T. Path P cannot intersect (out of K) with both of P 1 and P 2, because otherwise T contains a cycle. So, we may assume that P, P 1 and K form a path and d T (x,u) = P + P 1 + K. But d T (x,u) d, since d is the diameter of T. Therefore, d T (K,x) = P = d T (x,u) P 1 K d 2. But for every vertex x G, d G (K,x) d T (K,x). Also, d 2 t, since d t. 2 So K is a t-center of G, which is a contradiction. As a corollary, a tree is a t-star if and only if it has diameter at most t, because a tree has only one spanning tree. Now, for general graphs, if a graph is a t-star, then the diameter of the graph is at most t but there are graphs of diameter d that are not d-stars; for example, a cycle on eight vertices has diameter 4, but it is not a 4-star. When we examine a graphwhich is not at-star, we actually facethe treet-spanner problem. Before presenting a frequently used lemma, we give a definition to handle long paths. Definition A t midst M(P,t) of a path P from u to v is a subpath of P consisting exactly of either one vertex when t is odd, or a pair of adjacent in M vertices when t is even, such that d P (M,u) > t 1 2 and d P(M,v) > t 1 2. Obviously, a path P has a t-midst if and only if P t+1. There may be many t-midsts in a path but only if P = t+1, does P have a unique t-midst. Clearly, for any t-midst M, we see that M = E(M) = (t+1) mod 2. Lemma Let G be a graph and T a tree t-spanner of G. If M is a t-midst of a u, v-path P of T, then every u, v-path P of G contains a vertex whose distance from M in T is at most t 1 2. Proof. Since G contains a path that admits a t-midst, t is not zero. Let x = M, when t is odd, and x = E(M) when t is even. Consider the components of T\x. Note 13

19 u? w w M v W Figure1.1: LetW (solidcircle) bethevertices ofgwithindistance t 1 2 fromt-midst M in T. How are the components of T \M (dashed lines) related to the components of G\W? Here, t is odd. that when t is even, only two components are formed. Obviously, vertices u and v belong to different such components. Therefore, for any u, v-path P of G there is an edge ww in P such that w is in a different component than w (see figure 1.1). Since all the tree paths connecting vertices of different such components pass through x, it holds that d T (w,w ) = d T (w,m)+d T (M,w )+ M. But the tree distance between w and w can be at most t, therefore at least one of w or w is within distance t 1 2 from M (consider different cases when t is odd or even; note that when t is even the edge of M participates in the tree path between vertices w and w ). 1.5 Overview Our main goal is to study the set of tree t-spanners of G for every graph G and number t. The second chapter is on graphs that admit exactly one tree t-spanner. A characterization of graphs that admit a tree t-spanner (theorem 2.1.2), which is based on the notion of a t-star, provides intuition on the way tree t-spanners are formed in a graph and therefore hints towards properties of graphs that admit only one tree t-spanner. Trivially, a tree admits a unique tree t-spanner for each positive t, because a tree has only one spanning tree. Theorem affirms that a nontree graph G admits a unique tree t-spanner for at most one value of stretch factor t. A corollary of this theorem provides for each t a characterization of bipartite graphs that admit a unique tree t-spanner. One can efficiently recognize graphs that admit a unique tree 2-spanner but no 14

20 such algorithm is known, for t 3. At least, we prove that for each t there exist nontree graphs that admit a unique tree t-spanner. First, we present a sufficient condition for a graph to admit a unique tree t-spanner. Second, we provide a uniform way to construct an infinite family of graphs that admit a unique tree t-spanner, for each even value of stretch factor t. Also, for odd values there is another uniform way to construct such graphs. Structural differences between these families hint that the parity of t should be a major feature in any characterization of graphs that admit a unique tree t-spanner. Often though, if a graph admits a tree t-spanner, then it admits many tree t- spanners. How are the tree t-spanners of a graph related to each other? We examine this issue in the third chapter, where, for each t, we consider the union of any two tree t-spanners of a graph. A tree does not have any cycles but the union of two distinct trees on the same vertices does have cycles. The main tool for this study is theorem 3.1.2, which exhibits the interaction between tree t-spanners of a graph and cycles of the graph. Here, we pick induced cycles as the feature to describe relations between tree t-spanners of a graph and this leads to the conclusion that the t = 3 case is an exceptional one. Graphs being the union of two tree 2-spanners of a graph have a simple cycle structure: they do not contain any cycles of length greater than four and their only induced cycles are triangles; also, one can recognize such graphs efficiently. Then, we concentrate on the t = 3 case: let U 3 be the family of graphs being the union of two tree 3-spanners of a graph. Each graph in U 3 does not contain any induced cycle of odd length at least five. In contrast, for any even l 4 there is a graph in U 3 which contains an induced cycle of length l. Also, no graph in U 3 contains the complement of a cycle of length 5 or more as an induced subgraph. Thus, the Strong Perfect Graph Conjecture suggests that such graphs are perfect, which is proved in theorem When t 4, though, the union of two tree t-spanners of a graph may contain an induced odd cycle of any length or the complement of C 6 as an induced subgraph. Furthermore, we study the structure of induced cycles of graphs in U 3. Let G be 15

21 the union of two tree 3-spanners T 1 and T 2 of some graph. An induced cycle in G consists of edges of T 1 and T 2. We characterize the possible distributions of edges of T 1 and T 2 around such a cycle. Also, for every induced cycle C in G, there is a vertex of G which is adjacent in G to half of the vertices of C. Chapters four and five contain algorithmic results on the tree t-spanner problem. Graphs that admit a tree t-spanner of small diameter are the subject of chapter four. First, for a t-star G, recall that any shortest paths spanning tree to a t-center of G is a tree t-spanner of G. Second, according to theorem 4.1.1, if a graph admits a tree t-spanner of diameter at most t+1, then at least one of its shortest paths spanning trees is a tree t-spanner of the graph. Third, for each t 2, there are graphs which admit a tree t-spanner of diameter t+2 but none of their tree t-spanners is a shortest paths spanning tree. Therefore, should we expect that for each t there is an efficient algorithm to determine if a graph admits a tree t-spanner of diameter at most t+1? Theorem settles this question. Consider a graph G that admits a tree 3- spanner of diameter 4 and let K be its 4-center. Then, we know that G admits a tree 3-spanner T which is a shortest paths to K spanning tree of G. Here, K is just a vertex, so there is a vertex u such that all the edges of G incident to u are in T. We prove that finding the remaining edges of such a tree 3-spanner can be done efficiently. Though, for each t 4, the problem of determining if a graph admits a tree t-spanner of diameter at most t + 1 is an NP-complete problem, where we use a reduction from 3-SAT problem. Note that the situation for this problem from the complexity point of view is the same as the situation for the standard tree t-spanner problem, except for the t = 3 case. A promising attack to the tree 3-spanner problem is to find what prevents a graph from admitting a tree 3-spanner. So, we look for necessary conditions of tree 3-spanner admissibility; if such a condition does not hold for a graph, then the graph does not admit a tree 3-spanner. In chapter five, we introduce an integer linear programming formulation of the tree 3-spanner problem to join the power of such conditions. In this formulation, we consider an integer program for each graph, where we assign a 0-1 variable to each edge of the graph. So, we are interested in propositions which 16

22 can be used to efficiently generate linear constraints for such programs. First, we provide a procedure which outputs edges that have to be in every tree 3-spanner of the input graph. Second, the number of edges of a cycle in a graph that can be in a tree 3-spanner of the graph depends on the length of the cycle and some properties of the cycle. Third, for every vertex u of a graph G the number of edges incident to u that can be in a tree 3-spanner of the graph depends on the adjacencies between N G (u) and G\N G [u]. Definition describes for each graph an integer program that involves a variety of linear constraints, which are based upon propositions in this chapter. Then, a graph admits a tree 3-spanner if and only if the optimum value of its corresponding integer program is equal to n 1, where n is the number of vertices of the graph. This definition gives rise to a family of graphs for which there are certificates of tree 3-spanner inadmissibility, where we use the linear programming relaxations of these integer programs; to illustrate the use of such certificates, we give examples of graphs in this family. The thesis concludes with some guidelines for future research. 17

23 Chapter 2 Unique tree spanners Given a graph G and an integer t, we try to find at least one spanning tree of G which is a tree t-spanner of G. Are there graphs which admit only one tree t-spanner, though? The answer is trivially yes, since a tree G admits only one tree t-spanner for each positive t. But, in this case, notice that G contains only one spanning tree, as well. Also, a graph which is a tree plus one extra edge does not admit a unique tree t-spanner for any t. For each t, we present properties of graphs that admit a unique tree t-spanner and, also, we illustrate nontrivial examples of graphs that admit a unique tree t-spanner. It is necessary to make clear what we mean by a unique tree t-spanner. In particular, a spanning tree T of a graph G is the unique tree t-spanner of G if and only if T is a tree t-spanner of G and there is no subset of edges of G different than E(T) that forms a tree t-spanner of G. In contrast, note that if two isomorphic spanning trees of G are considered to be identical, then a cycle of length t+1 admits a unique tree t-spanner, which is a path of length t; for our purposes such a cycle admits t+1 tree t-spanners. A graph G admits a unique tree t-spanner if and only if each block of G admits a unique tree t-spanner. Therefore, we can concentrate on 2-connected graphs. 18

24 2.1 Tree t-spanner vs t-star In this section, we present a necessary and sufficient condition for a graph G to be tree t-spanner admissible, which exhibits possible ways to built a tree t-spanner of G. Therefore, this condition is helpful to come up with examples of graphs that admit a unique tree t-spanner. We present a lemma first. Lemma For any two trees T 1 and T 2, T 1 T 2 is a tree if and only if T 1 T 2 is a tree. Proof. Of course, if T 1 T 2 or T 2 T 1, then the lemma holds trivially. On one hand, assume that T = T 1 T 2 is a tree. First, since T is a tree, T and, therefore, F = T 1 T 2 is connected. Second, assume that F contains a cycle C. There are some edges of C which are not in T, because otherwise T has a cycle. Consider an edge e C, such that e T 1 (w.l.o.g. with respect to T 1 or T 2 ) but e T. Consider the maximum connected subgraph P of C, such that e E(P), E(P) E(C) E(T 1 ), and E(P) E(T) =. If E(P) = E(C), then P is a cycle in T 1, so P is a path. Let a and b be the endpoints of P. Note that the edges of C incident to a or b that are not in P belong to T 2, so both of vertices a and b belong to T. Since T is connected, there is a path Q(a,b) T. Also, recall that T T 1. Moreover, P(a,b) T 1, and P(a,b) is edge disjoint from T. So, paths P and Q form a cycle in T 1, which is a contradiction. Therefore, F is connected and does not contain any cycles. Hence, F is a tree. On the other hand, assume that T 1 T 2 is a tree. First, graph T 1 T 2 is a subset of this tree, so it does not contain any cycles. Second, if T 1 T 2 is not connected, then there is a path of T 1 connecting two components of T 1 T 2, and another path of T 2 connecting these two components. These two paths form a cycle in T 1 T 2, which 19

25 is a contradiction. By these two facts T 1 T 2 is a tree. Theorem A graph G admits a tree t-spanner if and only if G is a t-star, or G is the union of two induced proper subgraphs A and B of G that admit tree t- spanners T A and T B, respectively, such that a (t 1)-center of T A T B and a t-center of T A share a vertex. Proof. For the sufficient part, we prove that G is tree t-spanner admissible if either of the two cases of the theorem holds. So, for the first case, assume that G is a t-star. If G is the one vertex graph then G is trivially tree t-spanner admissible, otherwise let K be a t-center of that t-star. Any breadth first search tree of G starting from K is a spanning tree T of G such that d T (K,v) = d G (K,v) t, for all v in 2 G, since K is a t-center of G. Now, for every pair of vertices u,v in G: t t d T (u,v) d T (K,v)+d T (K,u)+ K + + K = t, 2 2 since K = t mod 2. Thus, T is a t-spanner of G. For the latter case, we prove that T = T A T B is a tree t-spanner of G. Here, t 1. Graph T A T B is a tree since it is a subgraph of tree T A and it is connected, because it admits a (t 1)-center. Therefore, by lemma 2.1.1, T is a tree. It remains to show that for every edge xy of G, d T (x,y) t. Since G is the union of graphs A and B, each edge of G belongs to A or B. If xy is an edge of A (similarly for B) then d T (x,y) = d TA (x,y) t, since T A is a tree t-spanner of A. For the necessary part, we prove the following equivalent proposition: If G is not a t-star and G admits a tree t-spanner, then the latter case of the theorem holds for G. If t = 0, then there is no graph which is not a 0-star but admits a tree 0- spanner. Thus, t 1. Let T be a tree t-spanner of G. Since T is a spanning tree of G, by lemma 1.4.6, the diameter d of T is strictly greater than t, i.e. d t+1. Thus, there is a path P 1 (u,v) in T of length d. Consider now the t-midst M of P 1 such that d P1 (u,m) = 1 + t 1 2, i.e. M is the closest to u t-midst of P 1. Since 1 An earlier version of this theorem appears in [30] 20

26 M = (t+1) mod 2 = (t 1) mod 2, a (t 1)-center of a (t 1)-star has the same size as M. Let S be the subtree of T induced by the vertices within distance t 1 2 from M in T. Thus, M is a (t 1)-center of S. We define graphs A and B. If t is even, let K be the vertex of M closest to u in P 1 and, if t is odd, let K be the subpath of P 1 consisting of vertex M, the neighbor of M in P 1 towards u and the edge between them. Let H 1 be the union of components Q of G\S such that every vertex of Q is within distance t from K in T. 2 We prove that H 1 contains vertex u. Let C be the component of G \ S that contains u and let x be any other arbitrary vertex of C. Of course, if u is the only vertex in C, then C is contained in H 1, since u is at distance t from K in T. 2 Towards a contradiction, assume that K is not contained in the path P T (x,v) of T from x to v. There are two cases for K. First, if K is a vertex, i.e. t is even, then M is not in P T (x,v) either (note that in this case K M), which implies that M is in P T (x,u) (Note that T is a tree). Second, if K is a pair of adjacent vertices, i.e. t is odd, then the vertex of K closest to u in P 1 is not in P T (x,v), because otherwise there would be a cycle in T, since K is not in P T (x,v) but K is in P 1. This implies that M is in P T (x,u). In both cases M is in P T (x,u). Also, M is a t-midst of P T (x,u), because x is not within distance t 1 from M in T, since x 2 is not in S. But there is a path between x and u in G that avoids S, since both of x and u are in C, which is a contradiction because of lemma Hence, K belongs to P T (x,v). Assume now that d T (x,k) > t 2. Since K belongs to P T(x,v), d T (x,v) = d T (x,k) + K + d T (K,v) > d T (u,k) + K +d T (K,v) = d (recall that d T (u,k) = t ), which is a contradiction, since d is the diameter of T. Therefore, all 2 x in C are at distance exactly t 2 from K, so C belongs to H 1. Next, H 1 does not contain vertex v, because d T (K,v) = M +d T (M,v) > M + t 1 = 2 t. So, there is at least one component of G\S which is not contained in 2 H 1. Let H 2 be the union of components of G\S that are not in H 1. Let A be the subgraph of G induced by H 1 S, and let B be the subgraph of G induced by H 2 S. Obviously, every vertex of G belongs to A or B and there is no edge of G joining a vertex in H 1 and a vertex in H 2, therefore G is the union of A 21

27 and B. Also, A and B are proper subgraphs of G, since H 2 and H 1 are not empty. Finally, let T A = T[A]. Note that T[S] is a tree and H 1 is not a cutset of T, so T A is a tree. For every vertex x in S d T (K,x) M +d T (M,x) M + t 1 2 = t 2. Also, every vertex in A \ S is at distance t 2 from K. So, K is a t-center of T A. Moreover, T A is a tree t-spanner of A, since for every pair of vertices in A the path between them in T A is the same as the path between them in T, where T is a tree t-spanner of G. Similarly, T B = T[B] is a tree t-spanner of B (T B is connected, since H 1 is not a cutset of T). By the definition of S, M is a (t 1)-center of T[S] = T A T B and M shares a vertex with K. Note that theorem subsumes various existing results. This theorem refers to all values of the stretch factor t. First, recall that a graph is tree 1-spanner admissible if and only if it is a tree. So, when t = 1, this theorem is a characterization of trees in terms of decomposition. Indeed, on one hand, if a tree T is not a 1-star, i.e. T is not the one vertex graph or the one edge graph, then T is the union of a tree T A and another tree T B, such that a 1-center of T A, i.e. T A is a 1-star (in this case, the 1- center of T A is T A itself), shares a vertex with a 0-center of T A T B, i.e. T A T B is just a vertex. On the other hand, the union of a tree T B and a one edge tree T A, i.e. T A has a 1-center, is a tree, whenever T A T B is exactly one vertex, i.e. T A T B has a 0- center. Second, the characterization of tree 2-spanner admissible graphs in[6, 10, 8] is an immediate corollary of theorem 2.1.2, when t = 2. Third, [8] contains a necessary condition for tree 3-spanner admissible graphs, which is subsumed here. In general, according to theorem 2.1.2, if a graph G admits a tree t-spanner and G is not a t-star, then G contains a subgraph, namely subgraph T A T B, which is a (t 1)-star and a cutset of G. 2.2 Stretch factor and uniqueness With respect to uniqueness, the case t = 0 is trivial, since the only tree 0-spanner admissible graph admits a unique tree 0-spanner. For t = 1, all the tree 1-spanner admissible graphs, i.e. all the trees, admit a unique tree 1-spanner. When t 2, 22