Boxicity of Leaf Powers
The boxicity of a graph , denoted as is defined as the minimum integer such that is an intersection graph of axis-parallel -dimensional boxes. A graph is a -leaf power if there exists a tree such that the leaves of the tree correspond to the vertices of and two vertices in are adjacent if and only if their corresponding leaves in are at a distance of at most . Leaf powers are a subclass of strongly chordal graphs and are used in the construction of phylogenetic trees in evolutionary biology. We show that for a -leaf power , . We also show the tightness of this bound by constructing a -leaf power with boxicity equal to . This result implies that there exists strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.
Key words: Boxicity, leaf powers, tree powers, strongly chordal graphs, interval graphs.
An axis-parallel -dimesional box, or -box in short, is the Cartesian product where each is an interval of the form on the real line. A 1-box is thus just a closed interval on the real line and a 2-box a rectangle in with its sides parallel to the axes. A graph is said to be an intersection graph of -boxes if there is a mapping that maps the vertices of to -boxes such that for any two vertices , . Then, is called a -box representation of . Thus interval graphs are exactly the intersection graphs of 1-boxes. Clearly, a graph that is an intersection graph of -boxes is also an intersection graph of boxes for any . The boxicity of a graph , denoted as , is the minimum integer such that is an intersection graph of -boxes.
Roberts gave an upper bound of for the boxicity of any graph on vertices and showed that the complete -partite graph with 2 vertices in each part achieves this boxicity. Boxicity has also been shown to have upper bounds in terms of other graph parameters such as the maximum degree and the treewidth. It was shown in  that for any graph on vertices and having maximum degree , . The same authors showed in  that . This result shows that the boxicity of any graph with bounded degree is bounded no matter how large the graph is.
The boxicity of several special classes of graphs have also been studied. Scheinerman  showed that outerplanar graphs have boxicity at most 2 while Thomassen  showed that every planar graph has boxicity at most 3. The boxicity of series-parallel graphs was studied in  and that of Halin graphs in .
Graphs which have no induced cycle of length at least 4 are called chordal graphs. Chordal graphs in general can have unbounded boxicity since there are split graphs (a subclass of chordal graphs) that have arbitrarily high boxicity . Strongly chordal graphs are chordal graphs with no induced trampoline (trampolines are also known as “sun graphs”). Several other characterizations of strongly chordal graphs can be found in , ,  and .
1.1 Leaf powers
A graph is said to be a -leaf power if there exists a tree and a correspondence between the vertices of and the leaves of such that two vertices in are adjacent if and only if the distance between their corresponding leaves in is at most . The tree is then called a -leaf root of . -leaf powers were introduced by Nishimura et. al. in relation to the phylogenetic reconstruction problem in computational biology. Characterization of 3-leaf powers and a linear time algorithm for their recognition was given in . Clearly, leaf powers are induced subgraphs of the powers of trees. Now, since trees are strongly chordal and any power of any strongly chordal graph is also strongly chordal (as shown in  and ), leaf powers are also strongly chordal graphs.
1.2 Our results
We show that the boxicity of any -leaf power is at most and also demonstrate the tightness of this bound by constructing -leaf powers that have boxicity equal to , for . The tightness result implies that strongly chordal graphs can have arbitrary boxicity. This is somewhat surprising because when we study the boxicity of strongly chordal graphs, it is tempting to conjecture that boxicity of any strongly chordal graph may be bounded above by some constant and small examples seem to confirm this conjecture. A subclass of strongly chordal graphs, called strictly chordal graphs, is studied in . The graphs in this class are shown to be 4-leaf powers in . Therefore strictly chordal graphs have boxicity at most 3.
2 Definitions and notations
We study only simple, undirected and finite graphs. Let denote a graph on vertex set and edge set . For any graph , the number of edges in it is denoted by . Thus, if is a path, denotes the length of the path. If is a tree that contains vertices and , then denotes the unique path in . For , let be the distance between and in . The -th power of a graph , denoted by , is the graph with vertex set and edge set .
A set of three independent vertices in a graph is said to form an asteroidal triple if for any , there exists a path between the two vertices in such that where denotes the set of vertices in . A graph is said to be asteroidal triple-free, or AT-free in short, if it does not contain any asteroidal triple.
Lemma 1 (Lekkerkerker and Boland)
A graph is an interval graph if and only if it is chordal and asteroidal triple-free.
If are graphs on the same vertex set , we denote by the graph on with edge set .
Lemma 2 (Roberts)
For any graph , if and only if there exists a collection of interval graphs such that .
A critical clique in a graph is a maximal clique such that every vertex in the clique has the same neighbourhood in . The critical clique graph of a graph , denoted as , is a graph in which there is a vertex for every critical clique of and two vertices in are adjacent if and only if the critical cliques corresponding to them in together induce a clique in .
For any graph , .
Since is an induced subgraph of , . Now suppose that is a vertex in and is the graph formed by adding a vertex to such that and . Since a -box representation for can be obtained from a -box representation for by extending to by defining , . Now since any graph can be obtained from by repeatedly performing this operation, .
A graph is a -Steiner power if there exists a tree , called the -Steiner root of with , and an injective map from to such that for , . Note that is induced in by the vertices in .
Lemma 4 (Dom et al.)
For , a graph is a -leaf power if and only if is a -Steiner power.
We first study the boxicity of tree powers and then deduce our results for leaf powers as corollaries.
3 Boxicity of tree powers
3.1 An upper bound
We show that if is any tree, boxicity of is at most .
Let be any tree. Fix some non-leaf vertex to be the root of the tree. Let be the number of leaves of the tree . Let be the leaves of in the order in which they appear in some depth-first traversal of starting from .
Define the ancestor relation on as follows: a vertex is said to be an ancestor of a vertex , denoted as , if . Similarly, we use the notation to denote the fact that is a descendant of , or in other words, is an ancestor of .
For any vertex , let be the parent of , i.e. the only ancestor of adjacent to it. Let . For any vertex , we define , and , for .
For any vertex , define to be the set of indices of leaves of that are descendants of , i.e., . Define and .
If , then .
. Hence the lemma follows.
If and , then either or .
Since the leaves were ordered in the sequence in which they appear in a depth-first traversal of from , for any vertex , the leaves in appear consecutively in the ordering . Since and , . This proves the lemma.
In order to show that , we construct interval graphs
such that . These
interval graphs are constructed as follows.
Construction of , :
Let be the interval assigned to vertex in , i.e., and . is defined as:
Note that from Lemma 5, since either or . Therefore is always a valid closed interval on the real line.
Construction of :
and where is defined as:
For , is a supergraph of .
Let . We will show that . Let be the path between and in . Since , . It is easy to see that there is exactly one vertex on such that and . Note that is the least common ancestor of and . Let and . Thus, and .
Let us assume without loss of generality that .
is a supergraph of .
Let . We have to show that . Let and let be the vertex on such that and (i.e., is the least common ancestor of and ). Let , and . We have and . Also, since , . Therefore, which means that . Thus, we have implying that .
If , then either or such that .
Let . Let and again let be the least common ancestor of and , i.e., is the vertex on such that and . Define and ; thus, . Since , we have .
If , consider the interval graph where . Now, let . Now, from (1), we get , that is to say . Thus, which means that .
If , then consider . From (1), we have , and therefore . Thus, .
Now, if , then and . This implies that . Similarly, if , then and implying that . In either case, , and so .
For any tree , , for .
If is a -leaf power, , for .
3.2 Tightness of the bound
Let the function be defined recursively as
, and for any ,
For any and , let be the tree shown in figure 1.
Let us prove this using induction on . It is easy to see that and (in vertices and form an asteroidal triple and therefore by Lemma 1, is not an interval graph). Let be a positive integer and assume that the statement of the lemma is true for any . We shall now prove by contradiction that . For ease of notation, let . If , then by Lemma 2, there exist interval graphs such that . Let . For each interval graph , choose an interval representation . For any and , let () denote the left (right) endpoint of its interval in . We define , i.e. the set of all vertices in the -th layer of . Let denote , the interval that corresponds to the vertex in . Note that, since , the vertices in layer of form a clique. Therefore, by Helly property, in the interval representation of each interval graph , the intervals corresponding to the vertices of layer 1 have a common intersection region. Let and denote the left and right endpoints respectively of this common intersection region in . That is, .
Since a vertex in , say , is not adjacent to any vertex in layer 1, for , there exists at least one interval graph such that is disjoint from the abovementioned common intersection region . Define , i.e., the collection of all interval graphs in which is not adjacent to at least one vertex in layer 1.
Also define , i.e., the set of all vertices in layer whose intervals are disjoint from in . Let us partition into two sets and .
Partition into two sets and such that and . Since , we encounter at least one of the following two cases. We will show that both the cases lead to contradictions.
Let us partition into sets where . Since , there exists an with . For a vertex , can be either to the left or to the right of in . Thus can be further partitioned into and where and . Since , we have or . Without loss of generality, let with . Also assume without loss of generality that . Since is adjacent to , we have . Also, by the definition of , . Therefore, contains both the points and , implying that it also contains . Thus, . Since , we know that for all , and therefore . This implies that , a contradiction.
For , let and let . Define . Note that both and exists since and thus . Let . Thus is a partition of into sets. Since , there exists such that . Now we partition into 4 sets namely,
Since , one of these 4 sets will have cardinality at least . Let this set be (the proof is similar for all the other cases). Thus contains vertices, which we will assume without loss of generality to be . Note that for any , and . Let . Now, since in any vertex in is adjacent to both and to all the vertices of layer 1, we have and . Since , contains the point . Similarly, contains the point . Thus, induces a clique in both and . Since is a universal vertex in , also induces a clique in both and . We claim that in , the induced subgraph of is isomorphic to . To see this, let . The isomorphism is given by the bijection where and . It can be easily verified that is an isomorphism from the graph induced in by to . Let
Since induced a clique in and , the induced subgraph on in is the same as the induced subgraph on in , i.e., is an induced subgraph of . Therefore, (from Lemma 2). But this contradicts the induction hypothesis.
We now construct a tree (see figure 2), for any and . Define .
We prove this by contradiction. Again, for ease of notation, let . Assume that . By Lemma 2, there exists a collection of interval graphs such that . Now for each interval graph , for , choose an interval representation . For a vertex , let () denote left(right) endpoint of its interval in . Let be the set of all vertices in the -th layer of .
For each vertex , since , there exists at least one interval graph in which . For each interval graph , we define . Note that is a partition of . We define a partition of into two sets and as follows. For any vertex , is in if the interval corresponding to is to the left of the interval corresponding to in , otherwise it is in . That is,