Treedepth Bounds in Linear Colorings

02/27/2018 ∙ by Jeremy Kun, et al. ∙ Google NC State University 0

Low-treedepth colorings are an important tool for algorithms that exploit structure in classes of bounded expansion; they guarantee subgraphs that use few colors are guaranteed to have bounded treedepth. These colorings have an implicit tradeoff between the total number of colors used and the treedepth bound, and prior empirical work suggests that the former dominates the run time of existing algorithms in practice. We introduce p-linear colorings as an alternative to the commonly used p-centered colorings. They can be efficiently computed in bounded expansion classes and use at most as many colors as p-centered colorings. Although a set of k<p colors from a p-centered coloring induces a subgraph of treedepth at most k, the same number of colors from a p-linear coloring may induce subgraphs of larger treedepth. A simple analysis of this treedepth bound shows it cannot exceed 2^k, but no graph class is known to have treedepth more than 2k. We establish polynomial upper bounds via constructive coloring algorithms in trees and intervals graphs, and conjecture that a polynomial relationship is in fact the worst case in general graphs. We also give a co-NP-completeness reduction for recognizing p-linear colorings and discuss ways to overcome this limitation in practice.

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1 Introduction

Algorithms for graph classes that exhibit bounded expansion structure boundedexpansion1 ; boundedexpansion2 ; boundedexpansion3 ; sparsity offer a promising framework for efficiently solving many NP-hard problems on real-world networks. The structural restrictions of bounded expansion, which allow for pockets of localized density in globally sparse graphs, are compatible with properties of many real-world networks such as clustering and heavy-tailed degree distributions. Moreover, multiple random graph models designed to mimic these properties have been proven to asymptotically almost surely belong to classes of bounded expansion demaine2014structural . From a theoretical perspective, graphs belonging to classes of bounded expansion can be characterized by low-treedepth colorings of bounded size, i.e. using only a small number of colors. Roughly speaking, a low-treedepth coloring is one in which the subgraphs induced on each small set of colors have small treedepth, a structural property stronger than treewidth. This definition naturally implies an algorithmic pipeline boundedexpansion2 ; demaine2014structural ; dvorak2013testing for classes of bounded expansion involving four stages: computing a low-treedepth coloring, using the coloring to decompose the graph into subgraphs of small treedepth, solving the problem efficiently on each such subgraph, and combining the subsolutions to construct a global solution. The complexities of algorithms using this paradigm often are of the form where is the coloring size and is the treedepth of the subgraphs.

A recent implementation concuss and experimental evaluation concuss_benchmarking of this pipeline has identified that the coloring size has a much larger effect on the run time than the treedepth in practice. Although graphs in classes of bounded expansion are guaranteed to admit colorings of constant size with respect to the number of vertices, the only known polynomial-time algorithms for computing these colorings are approximations sparsity . Consequently it is unclear to what extent our current coloring algorithms can be altered to reduce the coloring size. A more viable approach to improving the performance of the algorithmic pipeline without significant high-level changes would be to develop a new type of low-treedepth coloring that uses fewer colors but potentially has weaker guarantees about the treedepth of the subgraphs.

The traditional low-treedepth colorings for classes of bounded expansion are known as -centered colorings. This name stems from the property that on any subgraph , a -centered coloring either uses at least colors or is a centered coloring, which restricts the multiplicity of colors in induced subgraphs. In this paper we introduce an alternative that closely mirrors this paradigm but only extends the color multiplicity guarantees to path subgraphs. For this reason we refer to them as -linear colorings and linear colorings. We identify that -linear colorings share three important properties with -centered colorings that allow them to be used in the bounded expansion algorithmic pipeline.

  1. The minimum coloring size is constant in graphs of bounded expansion.

  2. A coloring of bounded size can be computed in polynomial time.

  3. Small sets of colors induce graphs of small treedepth.

The third of these properties is of particular interest, since understanding the tradeoffs between coloring size and treedepth in switching between -centered and -linear colorings fundamentally depends on bounding the maximum treedepth of a graph that admits a linear coloring with colors. Equivalently, we frame this problem as determining the gap between the minimum number of colors needed for a linear versus a centered coloring in any given graph. Using a grid minors approach, we prove that the minimum size of a centered coloring is polynomially bounded in the minimum size of a centered coloring. Because the “heavy machinery” of this approach likely does not give a tight bound, we give stronger upper bounds on the gap in trees and interval graphs and a matching lower bound for binary trees. Surprisingly, we also prove that some -linear colorings cannot be verified in polynomial time unless and discuss the practical implications of these findings. Some results in this paper appeared previously in WG 2018 lc_wg . This version adds a polynomial treedepth upper bound for general graphs, as well as tighter lower and upper bounds for trees.

2 Definitions and Background

In this section we detail the background and terminology necessary to understand -linear colorings.

2.1 Graph Terminology

We denote the vertices and edges of a graph as and , respectively, and assume all graphs are simple and undirected except where specifically noted otherwise. The open neighborhood of a vertex , denoted , is the set of vertices such that , while the closed neighborhood, is defined as . Vertex is an apex with respect to a subgraph if .

We say is a -path if for distinct and ; we will notate this as . Given disjoint paths and , the path is the concatenation of and if and are adjacent. A path is Hamiltonian with respect to subgraph if .

In a rooted tree , we let be the subtree of rooted at and the leaf paths of be the set of paths from a leaf of to . We label the levels of from bottom to top starting from 1; that is, if is the maximum distance from a leaf to the root then the root is the only vertex in level and level consists of all vertices whose parents are in level . Vertices and are unrelated in if is neither an ancestor nor a descendant of .

A coloring of a graph is a mapping of the vertices of to colors and has size . A coloring is proper if no pair of adjacent vertices have the same color. For any subgraph and color , if there is exactly one vertex such that we say appears uniquely in and is a center of . A subgraph with no unique color is said to be non-centered.

We use the notation to denote that form a partition of ; that is, and the sets are pairwise disjoint.

2.2 -Centered Colorings and Bounded Expansion

Definition 1

A -centered coloring of graph is a coloring such that for every connected subgraph , has a center or uses at least colors.

Nešetřil and Ossana de Mendez established that bounding the minimum size of a -centered coloring is a necessary and sufficient condition for a graph class to have bounded expansion.

Proposition 1 (boundedexpansion1 )

A class of graphs has bounded expansion iff there exists a function such that for all and all , admits a -centered coloring with colors.

There are varying methods to compute -centered colorings, such as transitive-fraternal augmentations boundedexpansion1 ; grohe2017deciding and generalized coloring numbers zhu2009colouring , we focus here on distance-truncated transitive-fraternal augmentations (DTFAs) FelixThesis , which iteratively augment the graph with additional edges to impose constraints on proper colorings. This linear time algorithm guarantees that after DTFA iterations, any proper coloring of the augmented graph is a -centered coloring whose size is bounded in classes of bounded expansion.

2.3 Centered Colorings and Treedepth

Note that if is a -centered coloring of and is a subgraph of whose vertices use at most colors in , must have a center. This relates -centered colorings to a more restricted class of graphs defined by centered colorings.

Definition 2

A centered coloring of graph is a coloring such that every connected subgraph has a center. The minimum size of a centered coloring of is denoted .

Note that a centered coloring is also proper, or else there would be a connected subgraph of size two with no center. Observe that if is the set of all centers of , then must either be empty or disconnected. This implies that if , then breaks into many components after only a few vertex deletions. This property is captured by treedepth decompositions.

Definition 3

A treedepth decomposition of graph is a rooted forest with the same vertex set as such that implies is an ancestor of in or vice versa. The depth of is the length of the longest path from a leaf of to the root of its component. The treedepth of , , is the minimum depth of a treedepth decomposition of .

Given a centered coloring of size , we can generate a treedepth decomposition of depth at most by choosing any center to be the root and setting the children of to be the roots of the treedepth decompositions of the components of . Likewise, given a treedepth decomposition of depth , we can generate a centered coloring using colors by bijectively assigning the colors to levels of the tree and coloring vertices according to their level. We refer to the colorings and decompositions resulting from these procedures as canonical; together they imply that the treedepth and centered coloring numbers are equal for all graphs.

3 -Linear and Linear Colorings

We introduce -linear colorings as an alternative to -centered colorings.

Definition 4

A -linear coloring is a coloring of a graph such that for every path111This includes non-induced paths. , either has a center or uses at least colors.

It is proven in FelixThesis that after performing DTFA iterations, any proper coloring of the augmented graph is a -linear coloring. This implies that -linear colorings indeed have constant size in bounded expansion classes and can be constructed in polynomial time (like -centered colorings).

In the interest of maintaining consistency with prior terminology, we define linear colorings analogously to centered colorings.

Definition 5

A linear coloring is a coloring of a graph such that every path has a center. The linear coloring number is the minimum number of colors needed for a linear coloring and is denoted .

Note that linear colorings must also be proper. A simple recursive argument shows that every path of length requires at least colors in a linear coloring; thus a graph of linear coloring number has no path of length . Because every depth-first search tree is a treedepth decomposition, , proving that small numbers of colors in -linear colorings induce graphs of bounded treedepth222This tightens a bound in FelixThesis from double to single exponential..

Our study of the divergence between linear and centered coloring numbers will naturally focus on linear colorings that are not also centered colorings. We say is a non-centered linear coloring (NCLC) of graph if contains a connected induced subgraph with no center. For NCLC , we say a connected induced subgraph is a witness to if is non-centered but every proper connected subgraph of has a center. For the sake of completeness, we prove in Lemma 1 that many simple graph classes do not admit NCLCs.

Lemma 1

If is a cograph, has maximum degree 2, or has independence number 2, any linear coloring of is also a centered coloring.

Proof

We analyze each graph class separately below.
Maximum degree 2: Let be a graph of maximum degree 2. Each connected induced subgraph of is either a path or a cycle, both of which have a Hamiltonian path. Thus every connected subgraph has a center, making any linear coloring centered.

Cographs: Let be an NCLC of cograph and be a witness to . If only contains one color, is an isolated vertex and the coloring is centered. Thus, we may assume has at least two colors. Because is a cograph, we can partition its vertices into nonempty sets such that is an edge in for all and . But since is proper, every pair of vertices with the same color must lie in the same set or . Since every color in appears at least twice, there are vertices and such that and but . But then form a path with no center and thus is not a linear coloring.

Independence number 2: Since independence number is hereditary, it is sufficient to show every connected graph of independence number 2 has a Hamiltonian path. We prove this by induction on the number of vertices, observing that an isolated vertex has a trivial Hamiltonian path. Let be a graph of independence number 2 and a vertex such that is connected, e.g., is a leaf in a minimum spanning tree of . If has a Hamiltonian cycle, then must have a Hamiltonian path. Otherwise, by the inductive hypothesis has a Hamiltonian path whose endpoints are some non-adjacent pair of vertices . Either is adjacent to one of , in which case has a Hamiltonian path, or form an independent set of size 3. ∎

The classes described in Lemma 1 are maximal in the sense that there are graphs with independence number 3 (graph described in Lemma 3) and binary trees (Lemma 4) that admit NCLCs.

4 Treedepth Lower Bounds

To understand the tradeoff between the number of colors and treedepth of small color sets when using -linear colorings in lieu of -centered colorings, it is important to know the maximum treedepth of a graph of fixed linear coloring number , . In Lemmas 3 and 4, we prove lower bounds on through explicit constructions of graph families. In order to show that these graphs have large treedepth, we first establish assumptions about the structure of treedepth decompositions that can be made without loss of generality.

Lemma 2

Let be a graph and such that is connected and with respect to some component , every vertex in is an apex of . Then for any treedepth decomposition of with depth , we can construct a treedepth decomposition such that:

  1. Each vertex in is an ancestor of every vertex in in

  2. For each pair of vertices or , is an ancestor of in iff it is an ancestor of in .

Proof

Let be a canonical centered coloring of with respect to . Let be a canonical treedepth decomposition with respect to ; if there are multiple vertices of unique color, prioritize removing those outside before members of , and then small colors over large colors, i.e., remove color 2 before color 5. Since is derived from a centered coloring with colors, its depth is at most , satisfying condition 1.

Condition 2 is satisfied as long each member of is removed in the construction of before any member of . Note that since contains apex vertices with respect to and every vertex satisfies , the removal of any vertex from cannot disconnect a previously connected component if has not been removed. Thus at any point in the algorithm before the removal of if a vertex in has a unique color in its remaining component , there must be another vertex in of unique color as well. Consequently, we will never be forced to remove any vertex of before .

To prove condition 3 is satisfied, observe that is an ancestor of in iff there is a connected subgraph containing and and no vertex with color smaller than . As stated previously, is a connected subgraph, which means that there is a subgraph witnessing this ancestor-descendant relationship between and such that if and if . Thus the relationships in are preserved in . ∎

Using Lemma 2, we now show that .

Lemma 3

There exists an infinite sequence of graphs such that

Proof

Define recursively such that is the empty graph and is a complete graph on vertices along with copies of for , call them , such that is an apex with respect to (Figure 1). We prove that and .

With respect to the linear coloring number, note that since the clique of size requires colors by Lemma 1. We prove the upper bound by induction on . The case of is trivial; assume it is true for . From the inductive hypothesis, we can assume each only requires colors for a linear coloring. Consider the coloring of such that and is a linear coloring of using colors . If is not a linear coloring, there is some path without a center. Since , must contain vertices from at least two s; each is a cut vertex, so cannot contain vertices from more than two s. However, , but , which means . Based on the symmetry of we can apply the same argument to the remaining colors, which means that no such non-centered path exists and is indeed a linear coloring of size .

With respect to the centered coloring number, by Lemma 2 there is an minimum-depth treedepth decomposition in which is an ancestor of . This implies there is a such that no vertex in shares a color in the canonical coloring with any of the vertices in the clique. Thus ; in the limit this recursion approaches . ∎

The graphs in Lemma 3 contain large cliques. We now show that this is not a necessary condition for the linear and centered coloring numbers to diverge.

Lemma 4

Let be the complete binary tree with levels. Then

Proof

Fix an integer and let be the smallest integer such that

(1)

Our proof proceeds by first constructing a coloring pattern of and then using to create a linear coloring for an arbitrarily large complete binary tree. Some vertices of will be left uncolored (we will call them local), while some vertices will be colored with one of the colors (we will call these colors global). Let be the sequence of all subsets of in order of nonincreasing size (in particular, and ) and let be such that . Note that such an index exists due to Equation (1):

and the fact that the sets are ordered in the nonincreasing order of their sizes. Furthermore, we have .

Let be an ordering of the leaves of corresponding to an in-order traversal. Consider an index . By construction, there exists a vertex at level that is the root of a subtree whose leaves are exactly for . We color the vertices of level by level with (global) colors of ; that is, we order the colors of arbitrarily and color level of with the -th color of for every . All remaining vertices of (that is, those that lie in none of the subtrees for ) remain local.

The following claim summarizes the properties of the above coloring.

Claim 1

For every path in that either

  • has both endpoints in a leaf or the root of the tree , or

  • does not contain a local vertex,

there exists a global color such appears uniquely on .

Proof

If a path does not contain a local vertex, then it is contained in a single tree . For such a path, the unique vertex on of maximum level is colored with a global color that appears uniquely on . Similarly, if is a leaf path in , then any globally colored vertex of the tree containing the leaf endpoint of satisfies the desired property. Otherwise, a path that has both endpoints in leaves of but contains a local vertex needs to start in a leaf of one subtree and end in a leaf of a different subtree . Then, observe that any (global) color of appears exactly once on .    

Let be the number of local vertices in the pattern . For an even integer , consider a coloring of defined as follows. Fix a palette of global colors and a palette of local colors. For every , the -th stripe consists of levels . In , such a stripe consists of copies of . Color every such copy using the pattern with global colors as the global colors of and color each local vertex with a different local color from the set if

is odd and from the set

if is even.

We claim that the above is a linear coloring of with colors. Consider a path in and let be the index of the highest stripe intersected by . By the choice of , intersects exactly one of the copies of in the -th stripe. If contains a leaf-to-leaf path in this copy, then Claim 1 asserts that contains a center in this copy (recall that every stripe uses a different set of global colors). Otherwise, intersects at most one copy of in every stripe. If intersects at least three stripes, then contains a root-to-leaf path in the single copy of intersected by at stripe , and we are again done by Claim 1. Similarly, Claim 1 finishes the proof if does not contain a local vertex at the -th stripe. Finally, in the remaining case intersects at most two stripes (the -th one and possibly the -th one) and contains a local vertex in the -th stripe. Since we used different set of local colors for odd and even stripes, any such local vertex in -th stripe is a center of .

Consequently, we have exhibited a linear coloring of with less than colors, where is defined as in Equation (1). If we let go to , then the ratio approaches . This ratio, in turn, approaches as due to the choice of at Equation (1). This finishes the proof of the lemma. ∎

Figure 1: Linear colorings of graph in Lemma 3

In Section 6 we show that the bound in Lemma 4 is tight for binary trees (Theorem 6.1). We conjecture that the construction in Lemma 3 is also tight for general graphs.

Conjecture 1

For any graph , .

While the exclusion of a path of length indicates , this nonetheless leaves a large gap between the upper and lower bounds on . To move towards a proof of Conjecture 1, we establish a polynomial upper bound on in general graphs in the next section (Theorem 5.1). Because this proof uses “heavy machinery”, we consider two restricted graph classes—namely, trees and interval graphs—in Sections 6 and 7 and give tighter upper bounds on for graphs in these classes.

5 Treedepth Upper Bounds on General Graphs

This section is devoted to proving a polynomial upper bound on .

Theorem 5.1

There exists a polynomial such that every graph satsifies .

Our starting point is the following theorem of Kawarabayashi and Rossman KawarabayashiR18 :

Theorem 5.2 (KawarabayashiR18 )

There is an absolute constant such that every graph of treedepth at least satisfies at least one of the following:

  1. the treewidth of is at least ;

  2. contains a complete binary tree of height as a minor;

  3. contains a path on vertices.

Assume that the treedepth of is at least . If contains a path on vertices (condition 3), then clearly . If contains a complete binary tree of height as a minor (condition 2), then also contains a subdivision of a complete binary tree of height as a subgraph. Since for any subgraph of , Theorem 6.1 asserts that . Thus, in the proof of Theorem 5.1, we are left with the case when has large treewidth.

Here, we use the celebrated grid minor theorem, with the best known bound due to Chuzhoy Chuzhoy16 .

Theorem 5.3 (Chuzhoy16 )

There is a polynomial such that every graph with treewidth at least contains a grid as a minor.

We slightly relax the notion of a grid minor to a -pseudogrid, defined as follows.

Definition 6

A graph contains a -pseudogrid if there exist two sequences of vertex-disjoint paths in , and such that

  • for every , the path is a concatenation of paths , , , , , , , in this order such that each path for is a subpath of (possibly consisting of a single vertex) and every path , does not contain any edge nor internal vertex on any path (we explicitly allow and to be paths of length );

  • a symmetric condition holds with the roles of and swapped.

In what follows, the paths , , , and are considered empty for pairs of indices not defined above.

Clearly, if contains a -grid as a minor, it contains a -pseudogrid: just let the paths follow the rows of the grid and the paths of follow the columns. To finish the proof of Theorem 5.1, it suffices to show the following technical result.

Lemma 5

If contains a -pseudogrid, then .

Proof

Fix a linear coloring of . Let be a -pseudogrid in . Let and similarly define . Let be the number of distinct colors uses on and similarly define . To prove the lemma, it suffices to show for any -pseudogrid in , . We shall prove it by induction over .

The statement is trivial for . For an inductive step, we proceed as follows. For a vertex , the grid coordinate of is if . A vertex is marginal if its grid coordinates satisfy , , , or . A color is infrequent on if it appears on , but there exists a family of at size at most such that every vertex with is either marginal or lies on one of the paths in . The definition of a color infrequent on is analogous.

For an inductive step, it suffices to show that there is always an infrequent color on or an infrequent color on . Indeed, assume that is infrequent on (the arguments for are symmetrical) and let be as in the above definition. Construct a -pseudogrid from as follows. Start with . First, delete from the first and last paths, and similarly for . Second, shorten every path by deleting the edges of and for and ; similarly shorten every path . Finally, delete all (shortened) paths of from , and delete a matching number of paths from . In this manner, we obtain a -pseudogrid such that and such that the color no longer appears on . Therefore, and . The inductive step follows.

In the remainder of the proof, assume that there is no infrequent color on nor an infrequent color on . We shall reach a contradiction by exhibiting a simple noncentered path .

We perform the following selecting and marking scheme. Initially, no vertex is selected and no path is marked. For every color that appears on , perform the following operation twice.

  1. Pick a vertex such that , is not marginal, and does not lie on a marked path . Let the grid coordinates of be .

  2. Select and mark all paths for and all paths for .

Now swap the roles of and and perform the above operation twice also for every color that appears on . In total, we select vertices. For every selected vertex we mark paths of and paths of . Since there is no infrequent color, there is always a vertex to choose at Step 1, as otherwise the so-far marked paths would witness infrequency of . Thus, the above selecting and marking scheme is well-defined.

Let be two distinct selected vertices and let and be their grid coordinates. By the above marking scheme, we have that

(2)

Consider now the following simple path . We start with being the concatenation of even-numbered paths without the prefixes and suffixes in the natural order, connected by paths for divisible by and by for (so that paths with are traversed forwards and paths with divisible by are traversed backwards). Then, for every selected with grid coordinates , we pick an even and modify locally to pass through . In the modification, we use only parts of paths , , , and . By Equation (2), two such modifications do not interfere with each other and no such modification interferes with the connections contained in paths and . Consequently, the final path is a simple path contained in that visits all selected vertices. Such a path does not contain a center, which is the desired contradiction. ∎

6 Treedepth Upper Bounds on Trees

Schäffer proved that there is a linear time algorithm for finding a minimum-sized centered coloring of a tree  schaeffer1989optimal . In this section we prove the following theorem by showing a correspondence between the centered coloring from Schäffer’s algorithm and colors on paths in any linear coloring of .

Theorem 6.1

Let be a tree of maximum degree , Then Schäffer’s algorithm finds a centered coloring of with size at most .

In particular, for trees of maximum degree we have , matching the lower bound of Lemma 4. We do not have any matching lower bound for larger . In fact, we conjecture that none exists, that is, the upper bound of Theorem 6.1 for is not tight.

Schäffer’s algorithm finds a particular centered coloring whose colors are ordered in a way that reflects their roles as centers. For this reason, the coloring is called a vertex ranking and the colors are referred to as ranks; it guarantees that in each subgraph, the vertex of maximum rank is also a center. We will use this terminology in this section to clearly distinguish between the ranks in the vertex ranking and colors in the linear coloring. Note that the canonical centered coloring of a treedepth decomposition is a vertex ranking if the colors are ranked decreasing from the root downwards, which implies that every centered coloring can be converted to a vertex ranking of the same size. Of central importance to Schäffer’s algorithm are what we will refer to as rank lists.

Definition 7

For a vertex ranking of tree , the rank list of , denoted , can be defined recursively as where is the vertex of maximum rank in .

Schäffer’s algorithm arbitrarily roots and builds the ranking from the leaves to the root of , computing the rank of each vertex from the rank lists of each of its children. For brevity, we denote for every in .

Proposition 2 (schaeffer1989optimal )

Let be a vertex ranking of produced by Schäffer’s algorithm and let be a vertex with children . If is the largest integer appearing on rank lists of at least two children of (or 0 if all such rank lists are pairwise disjoint) then is the smallest integer satisfying and .

We root at an arbitrary leaf of and let be a ranking output by Schaffers algorithm applied on (rooted) . With a vertex in we associate the following potential.

The following is immediate from Proposition 2:

Lemma 6

For every in with children , it holds that

Furthermore, the equality holds if and only if all rank lists are pairwise disjoint.

Let be a linear coloring of with colors. Our proof of Theorem 6.1 is based on tracking sets of colors of on paths terminating at the current vertex as Schäffer’s algorithm moves up the rooted tree. Given a path and a linear coloring of size , we say a color set is compatible with if both the following conditions are true:

  1. For every center , .

  2. For every color , there is a vertex such that .

In other words, a compatible set must not contain colors not found on , must contain each color appearing uniquely in , and may or may not contain any colors appearing multiple times on . For each , let be a set of sets defined recursively as follows. If is a leaf, . Otherwise, let be the children of , , and be an injective function such that

for all . Then . We start with the following straightforward observation.

Lemma 7

For every it holds that . Consequently, for every nonleaf , is a bijection between and .

We prove that the construction of preserves compatibility of sets.

Lemma 8

For all vertices and each , there is a corresponding path with as an endpoint such that is compatible with .

Proof

It is clear that the lemma holds at the leaves of , so we proceed by inductively showing the recursive step preserves the property. Observe that the path consisting of only is compatible with . For any , there is a child of such that is in . By the inductive hypothesis, there must be a path terminating at such that is compatible with . We claim that is compatible with . Since and each color appears the same number of times in and , it is only necessary to prove the requirements for compatibility are satified with respect to . Moreover, because appears at least once on it suffices to show that if , then appears multiple times on . By the definition of , implies and thus is not a center of . ∎

Define . We observe the following

Lemma 9

For any vertex with children , .

Proof

First, note that . Also, the lemma is straightforward for a leaf as then and . Assume then .

Recall that . Let be the set of all color sets that appear in exactly one and be those that occur in multiple ’s; we have . Note that for each , or else concatenating the corresponding compatible paths with creates a path with no center. Likewise, if there are distinct color sets and such that , then both belong to the same ; in particular, both belong to .

By the definition of , for each color set either or . In the latter case, there is a corresponding color set such that and . Also, from the discussion in the previous paragraph we infer that this latter case can only happen when . Hence,

We infer that

We conclude with the proof of Theorem 6.1.

Proof (Theorem 6.1)

For every leaf , we have . Lemmas 6 and 9 show inductively that for every . If is the size of the centered coloring output by Schäffer’s algorithm, then for the root of we have

Thus . ∎

7 Treedepth Upper Bounds on Interval Graphs

Because linear colorings are equivalent to centered colorings when restricted to paths, we turn our attention to the linear coloring numbers of “pathlike” graphs. We investigate a particular class of “pathlike” graphs in this section and prove a quadratic relationship between their centered and linear coloring numbers.

Definition 8

A graph is an interval graph if there is an injective mapping from to intervals on the real line such that iff and overlap.

We refer to the mapping as the interval representation of . Since the overlap between intervals and is independent of the interval representations of the other vertices, every subgraph of an interval graph is also an interval graph. The interval representation of implies a natural “left-to-right” layout that gives it the “pathlike” qualities, which are manifested in restrictions on the length of induced cycles (chordal) and paths between vertex triples (AT-free).

Definition 9

A graph is chordal if it has no induced cycles of length .

Definition 10

Vertices are an asteroidal triple (AT) if there exist -, -, and -paths , , and , respectively, such that . A graph with no AT is called AT-free.

Proposition 3 (lekkeikerker1962representation )

A graph is an interval graph iff is chordal and AT-free.

Intuitively, Definition 10 is a set of three vertices such that every pair is connected by a path that avoids the neighbors of the third. Roughly speaking, in the context of linear colorings, Proposition 3 indicates that if is a center of a “long” -path in , any vertex such that must have a neighbor on . We devote the rest of this section to proving Theorem 7.1.

Theorem 7.1

There exists a polynomial time algorithm that takes as input an interval graph and a linear coloring of with size and outputs a centered coloring of with size at most .

Our algorithm makes extensive use of the following well-known property of maximal cliques in interval graphs.

Proposition 4 (lekkeikerker1962representation )

If is an interval graph, its maximal cliques can be linearly ordered in polynomial time such that for each vertex , the cliques containing appear consecutively.

In particular, we identify a prevailing path in whose vertices “span” the maximal cliques and a prevailing subgraph that consists of the prevailing path as well as vertices in maximal cliques “between” consecutive vertices on the prevailing path. We will show that any linear coloring is a centered coloring when restricted to the prevailing subgraph and that after removing the prevailing subgraph, the remaining components each use fewer colors.

Let be an ordering of the maximal cliques of that satisfies Proposition 4. We say vertex is introduced in if but , and denote this as . Likewise, is forgotten in if but , and denote this as . The procedure for constructing a prevailing subgraph and prevailing path is described in Algorithm 1. This algorithm selects the vertex from the current maximal clique that is forgotten “last” and adds to the prevailing path and to the prevailing subgraph. We prove in Lemma 10 that if are a prevailing path and subgraph, the vertices in can be inserted between vertices of to form a Hamilt