Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SIAM J. COMPUT.
c
2011 Society for Industrial and Applied Mathematics
Vol. 40, No. 5, pp. 1292–1315
INTERVAL GRAPHS:
CANONICAL REPRESENTATIONS IN LOGSPACE
∗
JOHANNES K
¨
OBLER
†
, SEBASTIAN KUHNERT
†
, BASTIAN LAUBNER
†
, AND
OLEG VERBITSKY
‡
Abstract. We present a logspace algorithm for computing a canonical labeling, in fact, a
canonical interval represen tation, for interval graphs. To achieve this, we compute canonical in terval
representations of interval hypergraphs. This approach also yields a canonical labeling of convex
graphs. As a consequence, the isomorphism and automorphism problems for these graph classes are
solvable in logspace. For proper in t erval graphs we also design logspace algorithms computing their
canonical representations by proper and by unit in t erval systems.
Key words. graph isomorphism, graph canonization, logspace, interval graphs, int erval hyper-
graphs, convex graphs, proper interval graphs, unit interval graphs
AMS subject classifications. 05C60, 05C85
DOI. 10.1137/10080395X
1. Introduction. There has been persistent interest in the algorithmic aspects
of interval graphs in past decades, spurred much by their numerous applications;
see, e.g., [Gol04]. In 1976, Booth and Lueker presented the first recognition algo-
rithm for interval graphs [BL76] running in time linear in the number of vertices and
edges, which they followed up by a linear-time algorithm for interval graph isomor-
phism [LB79]. These algorithms are based on a special data structure called PQ-trees
that is used to enforce ordering constraints. By preprocessing the graph’s modular
decomposition tree, Hsu and Ma [HM99] later presented a simpler linear-time recog-
nition algorithm that avoids the use of PQ-trees. Habib et al. [HMP
+
00] achieve the
same time bound employing the lexicographic breadth-first search of Rose, Tarjan,
and Lueker [RTL76] in combination with smart pivoting. Parallel AC
2
recognition
and isomorphism algorithms were given by Klein in [Kle96].
All of the above algorithms have in common that they compute a perfect elim-
ination ordering (peo) of the graph’s vertices. This ordering has the property that
for every vertex, its neighborhood among its successors forms a clique. Fulkerson and
Gross [FG65] show that a graph has a peo if and only if it is chordal, and the above
methods determine whether a graph is an interval graph in linear time once a peo is
known.
Recognition of interval graphs in logspace follows from the results of Reif [Rei84]
and Reingold [Rei08]. In this article, we describe a logspace algorithm that, given
an interval graph G, constructs a canonical interval representation I
G
, i.e., G is
isomorphic to the intersection graph of I
G
, and isomorphic graphs G
1
∼
=
G
2
are
∗
Received b y the editors July 29, 2010; accepted for publication (in revised form) June 29, 2011;
published electronically September 20, 2011. A preliminary version of this paper appeared in Pro-
ceedings of the 37th International Colloquium on Automata, Languages and Programming, 2010.
http://www.siam.org/journals/sicomp/40-5/80395.html
†
Humboldt-Universit¨at, Institut f¨ur Informatik, Berlin, Germany (koebler@informatik.h u-
berlin.de, kuhnert@informatik.hu-berlin.de, laubner@informatik.hu-berlin.de). The second author’s
work was supported in part b y DFG grant KO 1053/7–1.
‡
Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences,
Lviv, Ukraine (v erbitsk@informatik.hu-berlin.de). This author’s work was supported in part by the
Alexander von Humboldt foundation.
1292
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INTERVAL GRAPHS: CANONICAL REPRESENTATIONS 1293
mapped to equal interval representations I
G
1
= I
G
2
. This, in particular, gives another
recognition algorithm and, moreover, implies that testing isomorphism of interval
graphs is also possible in logspace. Our methods are optimized for space complexity.
As such, our exposition neither relies on computing the graph’s peo nor uses transitive
orientation algorithms for comparability graphs as in [KVV85]. Instead, the basis of
our work is the observation of Laubner [Lau10] that in an interval graph, the set of
maximal cliques and a modular decomposition tree are definable in a certain logical
formalism, which makes these objects tractable in logarithmic space. We remark that
a logspace complexity bound implies an AC
1
bound, while logspace and linear time
results are incomparable (neither implies the other).
More specifically, we reduce canonization of interval graphs to that of interval
hypergraphs. We split interval hypergraphs into overlap components whose interval
representations are essentially unique; we show how to compute them canonically
using Reingold’s algorithm [Rei08]. After placing these components in a tree and
coloring them with their canonical interval representations, we apply Lindell’s tree
canonization algorithm [Lin92] and use its output to combine the canonical interval
representations of the components to one for the whole hypergraph. The tree used
in our algorithm is reminiscent of a PQ-tree in that it encodes all possible interval
representations. Our tree allows us to choose one of the representations in a canonical
way. This tree is constructible in logspace without the iterative refinement that is
inherent to the linear-time algorithms.
A hypergraph is an interval hypergraph if its vertices can be ordered so that
the vertices in each hyperedge are consecutive in this order. Switching to hyper-
graphs bears the advantage that these exhibit richer structure; this helps us to avoid
technical complications and to focus on the essence of the algorithm. Recognition
of interval hypergraphs is clearly equivalent to testing the so-called consecutive-ones
property: A matrix (which we interpret as the incidence matrix of a hypergraph) has
the consecutive-ones property for rows if its columns can be reordered such that the
ones in each row are consecutive. Testing for this property has complexity similar
to that of the recognition of interval graphs: Booth and Lueker gave a linear-time
algorithm that uses PQ-trees [BL76], which was later simplified by Hsu and Mc-
Connell [HM03]. Parallel AC
2
algorithms were given by Chen and Yesha [CY91]
and by Annexstein and Swaminathan [AS98]; they also follow from the parallel al-
gorithms for PQ-trees by Klein and Reif [KR88]. Our result implies that testing
the consecutive-ones property and finding an appropriate column permutation are in
logspace.
As another consequence of our logspace canonization of interval hypergraphs, we
show that convex graphs can be canonized in logspace. The isomorphism problem for
this class was previously known to be decidable by a parallel algorithm in AC
2
[Che96]
and by a sequential algorithm in linear time [Che99]. Convex graphs include bipartite
permutation graphs. The isomorphism problem for the latter class was only known
to be in AC
1
[CY93, YC96].
An interval graph is called proper if it admits an interval representation where no
interval is contained in another. Such representations can be found in linear time by
algorithms of Deng, Hell, and Huang [DHH96] and Hell, Shamir, and Sharan [HSS01].
An AC
2
algorithm is designed by Bang-Jensen, Huang, and Ibarra [BHI07]. We
show how to compute canonical proper interval representations in logspace, implying
also logspace recognition of proper interval graphs. Unit interval graphs are inter-
val graphs representable by systems of intervals of unit length. Any such graph is
obviously a proper interval graph, and the converse is also true by a classical result
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1294 K
¨
OBLER, KUHNERT, LAUBNER, AND VERBITSKY
of Roberts [Rob69]. Corneil et al. [CKN
+
95] show how to construct a unit interval
representation in linear time. Based on their methods, we observe that logspace is
sufficient to derive a unit interval representation from a proper one.
Finding logspace algorithms for the graph isomorphism problem of restricted
graph classes is an active research area. It was started by Lindell with his can-
onization algorithm for trees [Lin92]. In a series of results, Datta et al. generalize
this to planar graphs [DLN
+
09] (in fact, excluding one of K
5
or K
3,3
as minor is
sufficient [DNT
+
09]), whereas K¨obler and Kuhnert show the generalization to k-
trees [KK09]. The graph classes considered in these results have in common that
their clique size is bounded by a constant. To the best of our knowledge, our logspace
algorithm for interval graph isomorphism is the first for a natural class of graphs con-
taining cliques of arbitrary size. For all graph classes mentioned in this paragraph,
the isomorphism problem has a matching lower bound; i.e., it turns out to be logspace
complete.
Organization of the paper. Section 2 introduces some preliminaries, notably
the decomposition of interval hypergraphs into overlap components, and includes
a detailed overview of our algorithm in section 2.4. In section 3 we show how to
compute a canonical interval representation for a single overlap component in logspace.
Section 4 contains our main result: We give a logspace algorithm to obtain a canonical
interval representation of an arbitrary interval hypergraph. In section 5, we state our
results for interval graphs and convex graphs. Section 6 contains our algorithms for
proper and unit interval representations. In section 7 we summarize our results and
show that recognition and isomorphism testing of interval and convex graphs is hard
for logspace, thereby proving logspace completeness for these problems.
2. Definitions and basic facts. As usual, L is the class of all languages de-
cidable by Turing machines with a read-only input tape using only O(log n) bounded
space on the working tapes. FL is the class of all functions computable by such ma-
chines that additionally have a write-only output tape. For a set S,wedenoteits
cardinality by S.
2.1. Graphs and set systems. We write G
∼
=
H to say that G and H are
isomorphic graphs. The vertex set of a graph G is denoted by V (G). The set of
all vertices at distance at most 1 from a vertex v ∈ V (G) is called the (closed)
neighborhood of v and is denoted by N [v]. Note that v ∈ N [v]. We also use N[u, v]=
N[u]∩N[v] for the common neighborhood of two vertices u and v.IfN[u]=N [
v], we
call these vertices twins (note that only adjacent vertices can be twins according to our
terminology). We denote the degree of a vertex v ∈ V (G)asdeg(v)=N[v] \{v}.
Let F be a family of nonempty sets, which will also be called a set system.We
allow A = B for some A, B ∈F; i.e., F is a multiset whose elements are sets. The
support of F is defined by supp(F)=
X∈F
X.Aslot is an inclusion-maximal subset
S of supp(F) such that each set A ∈F contains either all of S or none of it.
The interse ction graph of F is the graph I(F) with vertex set F where A and B
are adjacent if and only if they have a nonempty intersection. Note that, if A = B,
these two vertices are twins in the intersection graph.
We consider intervals in the set of positive integers N, using the standard notation
[a, b]={i ∈ N | a ≤ i ≤ b}.Wesay[a
1
,b
1
] < [a
2
,b
2
]ifa
1
<a
2
or if a
1
= a
2
and
b
1
<b
2
. We extend this order to interval systems, i.e., multisets of intervals. For
interval systems I and J ,wewriteI < J if the smallest interval in the symmetric
difference of I and J (with due regard to the multiplicities) belongs to I.
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INTERVAL GRAPHS: CANONICAL REPRESENTATIONS 1295
AgraphG is an interval graph if it is isomorphic to the intersection graph of a
family of intervals I. This is equivalent to the standard definition of interval graphs as
intersection graphs of real intervals. Indeed, if we understand [a, b] as a real interval,
this does not change the intersection graph. On the other hand, given a finite system
of real intervals I, denote the set of all endpoints by P . Then the system of discrete
intervals induced by I on P has the same intersection graph. It remains to embed P
in N so that the order is preserved.
An isomorphism : V (G) →Ifrom G to I(I) is called an interval labeling of
G. The interval system I is called an interval representation of G and will also be
denoted by G
.
A labeling of a graph G is a bijection : V (G) →{1,...,V (G)}.Inthiscase
G
will denote the isomorphic image of G on the vertex set {1,...,V (G)}.The
canonical (interval) labeling pro blem for a class of graphs G consists of computing
for a given graph G ∈Gan (interval) labeling
G
such that G
G
= H
H
whenever
G
∼
=
H.Wecall
G
a canonical (interval) labeling and G
G
a canonical (interval)
representation of G. Note that solving the canonical interval labeling problem implies
solving the canonical labeling problem, as the intervals can be sorted and renamed.
2.2. Hypergraphs. We consider only hypergraphs (V,H) without isolated ver-
tices; i.e., the vertex set V is exactly supp(H). Hence, we often refrain from explicitly
mentioning V . In order to represent multiple hyperedges, we assign to each hyper-
edge H ∈Ha positive integer c(H) ≥ 1, called the multiplicity of H. It should be
stressed that, speaking of a hypergraph H, we will always suppose that H is a set
rather than a multiset. In other words, if hyperedges A ∈Hand B ∈Hare equal
as sets, they will be considered the same hyperedge (whose multiplicity can be more
than 1). An isomorphism from a hypergraph H to a hypergraph K is a bijection
φ: supp(H) → supp(K) such that
• H ∈Hif and only if φ(H) ∈Kfor every H ⊆ supp(H), and
• c(H)=c(φ(H)) for every H ∈H.
We say that two hyperedges A and B overlap and write A B
if A and B have
a nonempty intersection but neither of them includes the other. The overlap gr aph
O(H) is the subgraph of the intersection graph I(H) where the vertices corresponding
to the hyperedges A and B are adjacent if and only if they overlap.
Of course, O(H) can be disconnected even if I(H) is connected. A subset O of the
hyperedges of H corresponding to a connected component of O(H) will be referred
to as an overlap component of H. This is a subhypergraph of H and should not be
confused with the corresponding induced subgraph of O(H). Note that a hyperedge
of an overlap component inherits the multiplicity that it has in H.
If O and O
are different overlap components, then either every two hyperedges
A ∈Oand A
∈O
are disjoint or all hyperedges of one of the two components
are contained in a single slot of the other component. (This follows from a simple
observation that the conditions B ⊂ A, B B
,and¬(B
A)implythatB
⊂ A.)
This containment relation determines a tree-like decomposition of H into its overlap
components. In the case that O(H) is connected, H will be called an overlap-connected
hypergraph.
We call an interval system I an interval representation of a hypergraph H if
I viewed as a hypergraph is isomorphic to H. Hypergraphs having interval repre-
sentations are known in the literature as interval hypergraphs [BLS99, section 8.7].
Note that interval graphs are not just interval hypergraphs with hyperedges of size
2 (those are exactly unions of paths). This difference stems from the fact that inter-
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1296 K
¨
OBLER, KUHNERT, LAUBNER, AND VERBITSKY
vals correspond to the vertices of interval graphs and to the hyperedges of interval
hypergraphs.
Any isomorphism φ from H to I induces a labeling
φ
: H→Iof the hyperedges
in H with intervals from I where
φ
(H)=
φ(x) | x ∈ H
. We call a function
: H→Ian interval labeling of H if =
φ
for some φ. For another hypergraph
isomorphism ψ from H to I,wehave
φ
=
ψ
if and only if ψ
−1
φ maps every slot of
H onto itself. In other words, an interval labeling : H→Ispecifies an isomorphism
from H to I up to arbitrary rearrangements within slots.
Speaking of an interval representation I, we will suppose that supp(I)=[1,k],
where k = supp(I).Themapr(x)=k+1−x will be called the mirror reflection,and
the isomorphic interval system I
∗
= r(I) will be referred to as the mirror image of I.
An interval system I is mirror-symmetric if I
∗
= I. The mirror image of an interval
labeling : H→Iis the interval labeling
∗
: H→I
∗
defined as
∗
(A)=r((A)).
Lemma 2.1. Let H be an overlap-connected hypergraph with at least two hyper-
edges. Then H has either none or exactly two interval labelings, being mirror images
of each other.
Lemma 2.1 can be deduced from an equivalent statement in [CY91, Theorem 2].
For the reader’s convenience we here give a direct, independent proof.
Proof.Let: H→Ibe an interval labeling of H.SinceI and H are isomorphic,
O(I) is connected and I contains at least two intervals. The intervals I ∈Icontaining
1(resp.,supp(I)) will be called leftmost (resp., rightmost). Denote the longest
leftmost (resp., rightmost) interval in I by L (resp., R). Note that L = R or else I
would contain only one interval or O(I) would not be connected. Call a hyperedge
X ∈Hmarginal if the overlaps
X ∩ Y
Y ∈H,Y X
form a single inclusion chain.
It is not hard to see that (X) ∈{L, R} if and only if X is inclusion-maximal and
marginal. The latter conditions define an unordered pair of hyperedges, A and B,in
H, that does not depend on . Without loss of generality, suppose that (A)=L and
(B)=R. By definition,
∗
(B)=r(R).
Now consider any interval labeling
: H→I
mapping H to an arbitrary interval
system I
with supp(I
)=[1, supp(H)]. As we just observed, the leftmost interval
in I
equals either
(A)or
(B). Consider first the former case. We have
(A)=
[1, A]=L;thatis,
coincides with on A. Using induction on the distance d
between A and X in O(H), we prove that
and coincide on all X ∈H.Ifd =1,
then
(X)=(X) because both intervals must be equal to [A \ X +1, A ∪ X].
If d ≥ 2, let Z Y X be the terminal part of a shortest path from A to X in
O(H). By the induction hypothesis, we have
(Y )=(Y )=I and
(Z)=(Z)=J.
It suffices to show that the intervals I and J uniquely determine (X)and
(X)
and the determination rules for both are identical. Indeed, both (X)and
(X)
contain exactly one endpoint of I,whichissharedwithJ if and only if X and Z have
nonempty intersection. This determines the side of I where (X)and
(X)haveto
be attached. The exact position of (X)and
(X) is determined by the length of the
overlap with I, which is equal to X ∩ Y .Wehaveprovedthat
= .
In the case that
(B) is leftmost, we have
(B)=[1, B]=
∗
(B), and the same
argument shows that
=
∗
. Thus, there exists no interval labeling of H different
from and
∗
.
In section 3 we prove a constructive version of Lemma 2.1 (namely, Lemma 3.2),
allowing us to show that the unique pair of mutually reversed interval labelings is effi-
ciently computable. In fact, in section 3 we switch into another, equivalent language.
Givenanisomorphismφ from a hypergraph H to an interval system I,notethatφ
