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Unit Interval Graphs: Properties and Algorithms

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Unit Interval Graphs: Properties and Algorithms Unit Interval Graphs, Properties and Algorithms Sayyed Bashir Sadjad Hamid Zarrabi-Zadeh School of Computer Science University of Waterloo {bssadjad, hzarrabi}@uwaterloo.ca February 19, 2004 1 Introduction In this article we address some of the results on unit interval graphs. In short, a unit interval graph is an interval graph in which all intervals have the same length. These graphs have many applications in bioinformatics, databases, scheduling, measurement theory, etc. [KS96, BFMY83, PY79, Gol80]. We will study some of the problems that arises from such applications later in this article, after addressing properties of this class and available recognition algorithms. There are some other classes of graphs with different definitions which are in fact the same as the class of unit interval graphs. We introduce them in Section 2 and mention theorems that prove their equivalence. There are some efficient algorithms to check whether a graph is unit interval graph. Some of these algorithms are addressed in Section 3. A method for constructing efficient representations of unit interval graphs is given in Section 4. Finally, in Section 5 we investigate some problems on unit interval graphs that have faster algorithms than the general case. We also mention an interesting application of this class in bioinformatics in that section. 2 Definitions and Properties We start this section by giving three different definitions of three graph classes and then mention theorems showing that all these three classes are the same. When there are short proofs for the claims, we will give details of them, but for long proofs we only give references. Throughout this article we assume that the graphs are simple (with no loops or multiple edges) and connected with at least two vertices. To show an edge between vertices v and w we use (v, w) whether G is directed or undirected. So, when it is clear from the context that G is undirected, (v, w) and (w, v) are the same edges. Definition 2.1 An undirected graph G is an interval graph if there is a mapping f from vertices of G to a set I of real intervals such that (v, w) ∈ E(G) if and only if f(v) ∩ f(w) 6= φ. Such a set I is called an interval model for G. 1 v3 I3 I2 I1 I0 v0 v2 v1 Figure 1: A proper or unit interval graph cannot contain a K1,3. An interval graph is both chordal and co-comparability [Gol80]. On the other hand, if a graph is both chordal and co-comparability, then it is an interval graph. Therefore a nice characterization of the interval graphs is the following theorem: Theorem 2.2 (Golumbic [Gol80]) A graph G is an interval graph if and only if G is chordal and G¯ is a comparability graph. Definition 2.3 An interval graph G is a unit interval graph if there exists an interval model I for G in which all intervals are of equal size. In this case, I is called a unit interval model for G. Definition 2.4 An interval graph G is a proper interval graph if there exists an interval model I for G in which no interval properly contains another interval. In this case, I is called a proper interval model for G. It is not hard to show that the class of proper interval graphs is the same as the class of unit interval graphs. First of all, we show that a graph in any of these classes could not contain a K1,3 as an induced subgraph. Theorem 2.5 If G is a unit interval graph or a proper interval graph then G contains no K1,3 as an induced subgraph. Proof: The proof is given in Figure 1. In this figure, a K1,3 is shown on the left and an interval model for it given on the right (Ii is the interval of vi). As it is shown, I3 should be a proper subset of I0, so, K1,3 is not a proper interval graph. Moreover, it is clear that I0 and I3 cannot be of equal size. Therefore, K1,3 is not a unit interval graph. 2 The following theorem shows that this condition is also sufficient and implies that the class of unit interval graphs is the same as proper interval graphs. This theorem is from [BW99] but an extended version of it, which will be addressed later, is in [Gol80, BLS99, Rob69]. 2 Theorem 2.6 For a simple graph G the following three statements are equivalent: 1. G is a unit interval graph. 2. G is an interval graph with no K1,3 as induced subgraph. 3. G is a proper interval graph. Proof: The proof is by going from (2) to (3) and then from (3) to (1) (we know that (1) to (2) holds). To show that (2)⇒(3), we choose an interval model I for G, in which every vertex v of G is represented by an interval Iv of I. Starting from left to right, for any proper inclusion we move the interval with start point at right, to right without violating any edge condition. Suppose that there are two intervals Ix = [a, b] and Iy = [c, d] such that a < c ≤ d ≤ b (that is, Ix properly contains Iy). Since there is no induced K1,3 in G, in at least one of intervals [a, c] and [d, b] there is no endpoint of an interval Iz that does not intersect Iy. So, we can change the intervals Ix and Iy, without violating any edge condition, so that there is no proper inclusion. A similar idea works for going from (3) to (1). Here for each interval, we do some expansion and shrinking to make it unit and continue with the interval on the right. 2 Now, we define a third class of graphs which comes from measurement theories (attributed in [BW99] and [PD03] to [SS58]) and has close relations to unit and proper interval graphs. We have already seen a close connection between comparability graphs and interval graphs. We will see a closer relation between unit interval graphs and semiorders, which are a sub-class of partial orders. The following definition of semiorders is from [Gol80]. Definition 2.7 A partial order S = (V, P ) with elements V and a binary order relation P (on V ) is semiorder, if there is a function f : V → R from V to real numbers s.t. (u, v) ∈ P if and only if f(u) ≥ f(v) + 1. In the above definition, there could be any δ > 0 instead of 1 in the condition f(u) ≥ f(v) + 1. In Theorem 2.8, it is shown that all these definitions are equivalent. Historically, this is the root of unit interval graphs and Theorem 2.8 was proved in [Rob69] many years before [BW99]. Theorem 2.8 (Roberts [Rob69]) Let G = (V, E) be an undirected graph, then the following statements are equivalent: 1. There exists a function f : V → R s.t. (u, w) ∈ E(G) ⇔ |f(u) − f(w)| < 1. 2. There is a semiorder on S(V, P ) on the vertices of V such that (u, w) ∈ P ⇔ (u, w) 6∈ E(G). 3. G¯ is a comparability graph and every transitive orientation of G¯ is a semiorder. 4. G is an interval graph with no induced K1,3. 5. G is a proper interval graph. 3 z w w A. B. x y x y z Figure 2: A representation of Theorem 2.9 6. G is a unit interval graph. In [Rob69] it is mentioned that the (4)⇒(2) part was first observed by Marley through a personal communication with Roberts. Also, from historical point of view, the following theorem of Scott and Suppes [SS58] on semiorders is interesting since it is exactly the condition of having no K1,3 or induced Ck (for k ≥ 4) in the undirected complement of the relation. This is from [Gol80] but is attributed to [SS58]. Theorem 2.9 (Scott and Suppes [SS58]) A partial order relation S defined on V is a semiorder if the following conditions hold: 1. S is irreflexive. 2. (x,y) ∈ S and (z, w) ∈ S imply (x, w) ∈ S or (z,y) ∈ S. 3. (x,y) ∈ S and (y, z) ∈ S imply (x, w) ∈ S or (w, z) ∈ S. To see why these conditions are the same as the previous condition (interval graph with no K1,3), it is easy to check that condition (2) is equivalent to the condition that there is no Ck (for k ≥ 4) in G¯, where (u, w) is an edge in G = (V, E) if and only if neither (u, w) nor (w, u) is in S. This is shown in part A of Figure 2. Notice that in this figure, if there is no other directed edges (relation) between x,y,w and z, then in G there will be a C4 with no chord. On the other hand, notice that any other relation (i.e. other than xSy and zSw) then because of transitivity one of the white arrows should be present in S as well. So this is the necessary and sufficient condition for G to be chordal. Since S is a partial order, G¯ is both chordal and co-comparability, and hence G is interval. On the other hand, condition (3) is equivalent to the condition that there is no induced K1,3 in G (Figure 2 part B). Notice that since S is a partial order it is transitive so if xSy and ySz then xSz and the precondition of (3) means that x,y,z forms a clique in G¯ and the postcondition is that any vertex w in V should be connected to at least one of them in G¯ which means G does not contain any induced K1,3.
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