Tagged Probe Interval

A generalization of interval graph is introduced for cosmid contig mapping of DNA. A graph is a tagged probe interval graph if its vertex set can be partitioned into two subsets of probes and nonprobes, and a closed interval can be assigned to each vertex such that two vertices are adjacent if and only if at least one of them is a probe and one end of its corresponding interval is contained in the interval corresponding to the other vertex. We show that tagged probe interval graphs are weakly triangulated graphs, hence are perfect graphs. For a tagged probe interval graph with a given partition, we give a chordal completion that is consistent to any tagged interval completions with respect to the same vertex partition. Forbidden induced subgraph lists are given for trees with or without a given vertex partition. A heuristic that construct map candidates is given for cosmid contig mapping. 1 Cosmid Contig Mapping and Tagged Probe Interval Graphs All graphs in this paper are simple graphs without loops or parallel edges. Suppose F is a family of subsets of a set S. A graph G = (V;E) is the intersection graph of F if there is a bijection : V ! F such that for any u; v 2 V; uv 2 E if and only if (u) \ (v) 6= ;: A graph G = (V;E) is an interval graph if it is the intersection graphs of a family of intervals. Interval graphs have been used in physical mapping of DNA for reconstruction of relative positions of DNA fragments. In this paper, we introduce a generalization of interval graph for physical mapping of DNA. Structural properties related to algorithmic development are studied and used for constructing heuristic. In molecular biology, to replicate and study a certain contiguous stretch of DNA, DNA is cut into small fragments. Those fragments are replicated into clones. The goal of physical mapping is to reconstruct the relative positions of clones along original DNA. There are various techniques to determine if two clones overlap. Most of them involve obtaining ngerprints for clones, and determine that two clones overlap if their ngerprints are su ciently similar [2, 3, 6, 12]. Overlap relation of clones can be represented by an interval graph, where the vertices correspond to clones and two vertices are adjacent if their corresponding clones overlap. When complete overlap information is available and correct, there are e cient graph theory algorithms that produce a list of map candidates based on the interval graph model. On the other hand, reliable and complete overlap information is very costly and practically not available. Therefore, map construction could be more e cient if it can be done by using only a subset of overlap information. Di erent from ngerprinting, cosmid contig mapping generates overlap information by hybridization [4, 15]. In this process, a set of clones is placed on a lter for colony hybridization. The lter is probed with clones that have been radioactively labeled at their ends. We detect that two clones overlap if one of them is labeled and at least one end of this labeled clone is contained in the other clone. Interval graph model is not applicable for cosmid contig mapping. First, if only a subset of clones is selected to be labeled, we do not have complete overlap information among clones. Second, if an unlabeled clone is completely contained inside a labeled clone, their overlap cannot be detected. To model cosmid contig mapping and develop mathematical algorithms to construct maps, we introduce a generalization of interval graphs that capture the property of overlap information from cosmid contig mapping. Let G = (V;E) be a graph whose vertex set is V and edge set is E . Let V = fP;Ng be a partition of V . We call G a tagged probe interval graph with respect to P if there is a set of closed intervals I = fIvjv 2 V g such that vertices u and v are adjacent in G if and only if the following hold (1) Iu \ Iv 6= ;; (2) fu; vg \ P 6= ;; (3) flu; rug \ Iv 6= ; if u 2 P and v 2 N; where Iu = [lu; ru] : We call vertices in P probes and call vertices in N nonprobes. I is called a tagged (probe interval) representation of G with respect to the given partition. We call a graph G = (V;E) { 2 { a tagged probe interval graph if there is a partition V = fP;Ng such that G is a tagged probe interval graph with respect to P: When such a partition V = fP;Ng is given, we denote the graph by G = (V; P;E) to emphasize the given partition. If I is a tagged representation of G = (V; P;E) ; then the intersection graph of I is a tagged interval completion of G. We will denote Iu = [lu; ru]. We say interval Iv is properly contained in interval Iu if Iv Iu with lu 6= lv and ru 6= rv. In cosmid contig mapping, radioactively labeled clones correspond to vertices in P and the other clones correspond to vertices in N = V P . The task in cosmid contig mapping can now be interpreted as to test if a graph with a given partition is a tagged probe interval graph and to construct a tagged representation for G when G is a tagged probe interval graph. 2 Preliminaries LetG = (V;E) be a graph. For a subset S V; denoteG (S) the subgraph of G induced by S: Denote N (x) = fu 2 V jux 2 Eg the neighborhood of x 2 V and denote N [x] = N (x)[ fxg the closed neighborhood of x. If G = (V;E) is not an interval graph Booth76 G0 = (V;E [ E 0) is an interval graph, then G0 is an interval completion of G: We rst remark that it is possible that a graph can be a tagged probe interval graph for di erent partitions of probes and nonprobes. In practice, a partition, i.e., pre-selected probes and nonprobes, is often given by biologists. In the remainder of this paper, most of our results is given for graphs with given partitions. Tagged probe interval graphs introduced here can be viewed as a re nement of probe interval graphs. [15] and [14]. A graph G = (V;E) is a probe interval graph if the vertex set V can be partitioned into subsets P and N; and we can assign every vertex v 2 V an interval Iv such that uv 2 E if and only if Iu \ Iv 6= ; and fu; vg \ P 6= ;; where I is called a probe (interval) representation of G with respect to P: Probe interval graphs were also introduced for the cosmid contig mapping. If we assume that probes are not only labeled on their two ends, Booth76 also on their interior points, then probe interval graphs can be used to represent overlap information obtained. An interval graph is both a tagged probe interval graph and a probe interval graph with respect to P = V . A tagged probe interval graph or a probe interval graph may not be an interval graph. For example, it is well known that chordless cycle C4 of length four is not an interval graph. However, C4 is a tagged probe interval graph as well as a probe interval graph. On the other hand, there is no containment relation between tagged probe interval graphs and probe interval graphs. There are graphs that are probe interval graphs Booth76 not tagged probe interval graphs with respect to any possible partition. There are also graphs that are tagged probe interval graphs Booth76 not probe interval graphs with respect to any possible partition. The following lemma summarizes some observations which will be used in the remainder of this paper. { 3 { Lemma 2.1 Suppose G = (V; P;E) is a tagged probe interval graph and I = fIvjv 2 V g is a representation of G with respect to the given partition. (1) G (P ) is an interval graph. In particular, if P = V then G is an interval graph; (2) G (V P ) is an independent set; (3) If H V , then G (H) is also a tagged probe interval graph; (4) If u 2 P; v 2 N and uv 62 E, then Iv not properly contained in Iu implies that Iu \ Iv = ;; (5) If jfu; vg \ P j = 1 and uv = 2 E imply Iu \ Iv = ; for any u; v 2 V; then G is a probe interval graph with respect to P and I is a probe representation of G with respect to P: Proof. (1) (4) follow directly from the de nition. Condition (5) implies that uv 2 E if and only if at least one of u; v is a probe and Iu \ Iv 6= ;: Hence G is a probe interval graph and the lemma follows. 2 Suppose G = (V; P;E) is a tagged probe interval graph. We call two nonadjacent vertices v1; v2 an enhanceable pair if at least one of v1; v2 is in N and there are two nonadjacent vertices u1; u2 2 P such that uivj 2 E (i; j = 1; 2). The enhancement of G is obtained by adding an edge between every enhanceable pair. Denote the enhancement of G by G = (V; P;E [ E ) where E contains all new edges between enhanceable pairs. Edges in E are called enhanced edges. It is easy to see that if G = (V; P;E) is a tagged probe interval graph with I = fIvjv 2 V g a tagged representation, and G = (V; P;E [ E ) is the enhancement of G , then uv 2 E [ E implies that Iu \ Iv 6= ;. Theorem 2.2 Suppose G = (V; P;E) is a tagged probe interval graph. If every nonprobe of G is incident to an enhanced edge, then G0 = V; P;E [ E 0 is a probe interval graph where E 0 contains all enhanced edges that has a probe as one end. Proof. Let I = fIuju 2 V g be a tagged representation of G and let u 2 N be a nonprobe. Since u is incident to an enhanced edge, there are x; y 2 P such that xy = 2 E; xu 2 E; yu 2 E: Therefore Ix \ Iy = ;, Iu \ Ix 6= ;, and Iu \ Iy 6= ;. If a probe w 2 P is such that Iu \ Iw 6= ; and Iu does not contain any end point of Iw; then Iu is properly contained in Iw and uw = 2 E. It follows that Iw \ Ix 6= ;; Iw\ Iy 6= ; and uw 2 E 0. Therefore for any u; v 2 V , uv 2 E [E 0 if and only if Iu \ Iv 6= ; and fu; vg \ P 6= ;: By de nition, G0 is a probe interval graph. 2 To end this section, we present a general frame of classi cation of graphs that has tagged probe interval graphs and probe interval graphs as special cases. A graph G = (V;E) is a split graph if there is a partition V = fV1; V2g such that G (V1) is a clique and G (V2) is an independent set. We call a graph G = (V;E) a property-Q split graph with respect to V 0 for some subset V 0 V if G (V 0) is a graph of property Q and G (V V 0) is an independent set. Various classes of graphs in the literature can be considered as special cases of this generalization of split graphs. For example, apex graphs, i.e., planar graphs with an extra { 4 { vertex are special cases of planar split graphs according to our de nition here. Graphs that can be partitioned into a split graph and an independent set (or a clique) can be viewed as split split graphs, which are studied in [9]. In the same spirit, we call a graph G = (V;E) an interval split graph if there is a partition of V = fV1; V2g such that G (V1) is an interval graph and G (V2) is an independent set. Tagged probe interval graphs and probe interval graphs are special cases of interval split graphs. In the remainder of this paper, we call a graph G = (V; P;E) an interval split graph with respect to P when a partition V = fP;Ng is given and G (P ) is an interval graph and G (N) is an independent set. 3 Perfectness of Tagged Probe Interval Graphs In this section, we show that tagged probe interval graphs are perfect graphs. Given graph G = (V;E) ; the clique number ! (G) is the number of vertices in a maximum complete subgraph of G. The chromatic number (G) of G is the minimum number of independent sets needed to cover all vertices of G: G is perfect if ! (H) = (H) for all induced subgraphs H of G: Perfect graphs are important in combinatorial optimization and algorithmic theory. Interval graphs are well known perfect graphs. A graph is weakly triangulated if both G and Gc; the complement of G; do not have any induced subgraph isomorphic to a chordless cycle of at least ve vertices. Hayword [8] showed that weakly triangulated graphs are prefect graphs. The following was shown in [11]. Theorem 3.1 [11] Probe interval graphs are weakly triangulated. In the following, we show that, similar to probe interval graphs, tagged probe interval graphs are weakly triangulated graphs. Lemma 3.2 A chordless cycle Ck (k 5) is not a tagged probe interval graph. Proof. Let Ck = (V;E) (k 5) be a chordless cycle. Suppose on the contrary that fIvjv 2 V g is a tagged representation of Ck (k 5) with respect to some P V . Then V 6= P since Ck is not an interval graph. Suppose u; v are a pair of none adjacent vertices such that u 2 P and v 2 N . By (2) of Lemma 2.1, the two vertices adjacent to v in Ck, say x and y, must be probes. Since k 5 and uv = 2 E, it follows that fx; yg 6 N [u] and u 6= x; y. By (1) of Lemma 2.1, either Iu \ Ix = ; or Iu \ Iy = ;: Therefore Iv cannot be properly contained in Iu: Then uv = 2 E implies that Iu \ Iv = ;. By (5) of Lemma 2.1 Ck would be a probe interval graph, contradicting Theorem 3.1. 2 Lemma 3.3 The complement of any chordless cycle Ck (k 5) is not a tagged probe interval graph. { 5 { Proof. Let Cc k = (V;E) be the complement of Ck: Since Cc 5 = C5, Cc 5 is not a tagged probe interval graph with respect to any P V by Lemma 3.2. For k 6, assume on the contrary that I = fIvjv 2 V g is a tagged representation of Cc k with respect to some P V . Since Cc k has induced subgraphs isomorphic to C4, by (1) of Lemma 2.1, P 6= V . Similar to the proof of Lemma 3.2, let u; v be a pair of nonadjacent vertices of Cc k such that u 2 P and v 2 N . In the representation Iv cannot be properly contained in Iu for otherwise any vertex adjacent to v is a probe and hence adjacent to u by (1) of Lemma 2.1. It follows from uv = 2 E that Iu\ Iv = ;. By (5) of Lemma 2.1, Cc k (k 6) would be a probe interval graph, contradicting Theorem 3.1. 2 Theorem 3.4 Tagged probe interval graphs are weakly triangulated. Proof. By (3) of Lemma 2.1 an induced subgraph of a tagged probe interval graph is also a tagged probe interval graph. Therefore by Lemmas 3.2 and 3.3, a tagged probe interval graph cannot contain any induced subgraph isomorphic to Ck or Cc k (k 5) : 2 Corollary 3.5 Tagged probe interval graphs are perfect. 4 Chordal Completion of Tagged Probe Interval Graphs A graph is chordal or triangulated if it contains no induced subgraph isomorphic to Ck for any k 4. If G = (V;E) is not a chordal graph Booth76 H = (V;E [ E 0) is a chordal graph, then H is a chordal completion of G: If H has minimum number of edges among all chordal completion of G; then H is called a minimum chordal completion of G: In this section, for a tagged probe interval graph G with a given partition, we show that the enhancement of G is a chordal completion of G that is contained in any tagged interval completion of G with respect to the given partition. Theorem 4.1 Suppose that G = (V; P;E [ E ) is the enhancement of a tagged probe interval graph G = (V; P;E). Then G is a subgraph of any tagged interval completion of G with respect to P: Proof. Suppose that GI is a tagged interval completion of G. Let I = fIuju 2 V g be an interval representation of GI . Clearly every edge of G is an edge of GI . We only need to show that every enhanced edge is also an edge of GI . Suppose u1; u2 is an enhanceable pair of G: Then there are probes v1; v2 such that v1v2 = 2 E and uivj 2 E (i = 1; 2 and j = 1; 2). Hence Iv1 \ Iv2 = ;, Iui \ Ivj 6= ; (i = 1; 2 and j = 1; 2). It follows that Iu1 \ Iu2 6= ;: Therefore u1u2 is an edge of GI : 2 Lemma 4.2 Let G = (V; P;E) be a tagged probe interval graph with a tagged representation I = fIvjv 2 V g. (1) Suppose u1; u2; w 2 V are such that Iu1 Iw or Iu2 Iw. If u1u2 is an enhanced edge, then either uiw 2 E or uiw is an enhanced edge for i = 1; 2. { 6 { (2) Suppose Iu1 ; Iu2; Ip; Iv1; Iv2 2 I are such that ; 6= Iu1\Iu2 Iv1\Iv2: Then Ip\ Iui 6= ; for i = 1; 2 implies that Ip \ Ivi 6= ; for i = 1; 2. Proof. If u1; u2 is an enhanceable pair, there exist two nonadjacent probes p1; p2 such that Iui \ Ipj 6= ; (i = 1; 2 and j = 1; 2) : If Iu1 Iw or Iu2 Iw then Iw \ Ipj 6= ; (j = 1; 2). Therefore (1) follows. For (2), since intervals satisfy Helly property, Ip\Iv1\Iv2 Ip\ Iu1\ Iu2 6= ;. Therefore, Ip \ Ivi 6= ; and the lemma follows. 2 Theorem 4.3 Let G = (V; P;E) be a tagged probe interval graph and let E be the set of enhanced edges. Then G = (V; P;E [ E ) is a chordal graph. Proof. Let I = fIvjv 2 V g be a tagged representation of G. Let C be a cycle of G with vertices v1; ; vk (k 4) and edges vkv1 and vivi+1 (i = 1; :::; k 1). Let intervals corresponding to vi be Ii = [li; ri] (i = 1; :::; k). Without loss of generality, assume that rk = max frjj1 j kg and that rk rk 1 r1: Suppose on the contrary that C has no chords in G . Then v1vk 1 62 E, vk 2vk 62 E and v2vk 62 E. First consider the case that Ik 1 Ik: Note that Ik 2 \ Ik Ik 2 \ Ik 1 6= ;: Since rk rk 2; we have vk 2 2 N , for otherwise vk 2vk would be a chord by Ik 2 \ Ik 6= ;. If vk 1 2 N , then vk 1vk 2 2 E implies vk 2vk 2 E by (1) of Lemma 4.2, a contradiction. Therefore vk 1 2 P . If Ik 2 Ik, then vk 3 2 P for otherwise we would have that vk 3vk 2 2 E , and so vk 2vk 2 E by (1) of Lemma 4.2. Now since Ik 3 \ Ik 6= ; and rk 3 rk; vk 3vk 2 E. So k = 4 and vk 3 = v1. Hence, vk 2 and vk are each adjacent to two nonadjacent probes v1 and vk 1, which implies that vk 2vk 2 E . This contradicts our assumption on C, so Ik 2 6 Ik and lk 2 Ik 2. Hence, vk 2 N for otherwise we would have vk 2vk 2 E since Ik 2 \ Ik 6= ;. Suppose I1 \ Ik 1 6= ;. Then v1 2 N and I1 Ik 1 for otherwise v1vk 1 2 E. Hence, v1vk 2 E , which implies that v1vk 1 2 E by (1) of Lemma 4.2, a contradiction. Therefore I1 \ Ik 1 = ;, and so r1 < lk 1. Note that since Ik 2 6 Ik, Ik 2 \ Ik 1 6= ; and r1 < lk 1, we have r1 2 Ik 2 and ; 6= I1 \ Ik Ik 2 \ Ik. If v1 2 P , then vk 2v1 2 E since r1 2 Ik 2, and so k = 4. Since vk 2 and vk are each adjacent to two nonadjacent probes v1 and vk 1 , so vk 2vk 2 E , a contradiction. If v1 2 N , then v1vk 2 E would lead to vk 2vk 2 E by (2) of Lemma 4.2, again a contradiction. Combining above we have that if Ik 1 Ik then C has a chord. Next consider the case that Ik 1 6 Ik. Ik 1\ I1 6= ; since r1 rk 1 rk and I1\ Ik 6= ;. If I1 Ik 1, then v1 2 N since v1vk 1 = 2 E. It follows that vk 2 P and v2 2 P for otherwise we would have v1vk 1 2 E by (1) of Lemma 4.2. On the other hand, since I2 \ Ik 1 6= ;, it follows from Ik \ Ik 1 6= ; and v2, vk are two nonadjacent probes that v2vk 1 2 E. Thus v1 and vk 1 are each adjacent to two nonadjacent probes v2 and vk, and so v1vk 1 2 E , a contradiction. Therefore, I1 6 Ik 1 . Similarly, Ik 1 6 I1. Hence v1; vk 1 2 N and Ik 1 Ik I1 Ik. Now vk 2 P for otherwise since ; 6= I1 \ Ik Ik 1 \ Ik, v1vk 2 E would imply that v1vk 1 2 E by (2) of Lemma 4.2. { 7 { If Ik 2 \ Ik 6= ;, then vk 2 2 N and Ik 2 Ik since vk 2vk = 2 E. This is impossible since by (1) of Lemma 4.2, vk 2vk 1 2 E would imply vk 2vk 2 E . Therefore Ik 2 \ Ik = ;. It follows that Ik 2 \ I1 = Ik 2 \ (I1 Ik) Ik 2 \ (Ik 1 Ik) = Ik 2 \ Ik 1 6= ;. If vk 2 2 P , then since I1 \ Ik 2 6= ;, it follows from I1 \ Ik 6= ; and vk 2, vk are two nonadjacent probes that v1vk 2 2 E. Thus v1 and vk 1 are each adjacent to two nonadjacent probes vk 2 and vk, and so v1vk 1 2 E , a contradiction. Hence vk 2 2 N . Then, since Ik 2 \ I1 Ik 2 \ Ik 1 6= ;, vk 2vk 1 2 E would imply that v1vk 1 2 E by (2) of Lemma 4.2, again a contradiction. Therefore C must have a chord in G : 2 Notice that the enhancement G of a tagged probe interval graph G may not always be a minimum chordal completion of G. Furthermore, G may not be necessarily an interval graph yet. 5 Characterization of Cycle-Free Tagged Probe Interval Graphs Since tagged probe interval graphs are closed for the operation of taking induced subgraphs, tagged probe interval graphs can be characterized by a list of forbidden induced subgraphs. Similarly, tagged probe interval graphs can also be characterized by a list of forbidden induced subgraphs where every graph in the list has prescribed partition of probes and nonprobes. In the following, we present such lists for trees. Given a graph G = (V;E) ; a subset fv1; v2; v3g V is an asteroidal triple if there are paths Pij from vi to vj such that Pij \ N (vk) = ; for i; j; k = 1; 2; 3: The following is well-known: Theorem 5.1 [10] A chordal graph is an interval graph if and only if it has no asteroidal triple. Lemma 5.2 Let G = (V; P;E) be an interval split graph. If G contains an asteroid triple of three probes, then G is not a tagged probe interval graph with respect to P . Proof. Let x; y; z be an asteroid triple of G such that fx; y; zg P . Let Pxy be a path between x and y such that Pxy \ N (z) = ;. Let Pxz ; Pyz be similarly de ned. Suppose on the contrary that G = (V; P;E) is a tagged probe interval graph with a representation I = fIvjv 2 V g. Let H = (V; P;EH) be the intersection graph of I . Since H is an interval graph, fx; y; zg is not an asteroidal triple of H: Without loss of generality, suppose uz 2 EH for some vertices u on the path Pxy . Thus u 2 N and Iu Iz for otherwise uz 2 E. Note that since x; y are probes, u 6= x; y; let w be a vertex on Pxy next to u: Then w 2 P and Iw \ Iu 6= ;: Hence Iw \ Iz 6= ;; and so wz 2 E; which contradicts that z is not adjacent to any vertex on Pxy in G. 2