Chordal Probe Graphs (extended abstract)

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1 Chordal Probe Graphs (extended abstract) Martin Charles Golumbic Marina Lipshteyn Abstract. In this paper, we introduce the class of chordal probe graphs which are a generalization of both interval probe graphs and chordal graphs. A graph G is chordal probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. We show that a chordal probe graph may contain neither an odd-length chordless cycle nor the complement of a chordless cycle, and we present the complete heirarchy with separating examples for these classes. We give polynomial time recognition algorithms for the subfamily of chordal probe graphs which are also weakly chordal, first in the case of a fixed given partition of the vertices into probes and non-probes, and second in the more general case where no partition is given. 1 Introduction Let G = (V, E) be a finite, undirected, simple graph (i.e., without self-loops and parallel edges). The complement G = (V,E) of G has the same set of vertices and edge set defined by E = {(x,y) x,y V and x y and (x,y) / E}. Given a subset X V, the subgraph induced by X is G X = (V (G X ),E(G X )), where E(G X ) = {(v,w) E v X and w X} and V (G X ) = X. A set X V is an independent set or a stable set in G if for all u,v X, (u,v) / E. A set Y V is a clique in G if for all u,v Y, u v, (u,v) E. If Y = V, then G is a complete graph. A sequence [v 1,...,v k ] of distinct vertices is a path in G if (v 1,v 2 ),..., (v k 1,v k ) E. These edges are called the edges of the path. The length of the path is the number k 1 of its edges. A closed path [v 1,...,v k,v 1 ] is called a cycle if in addition (v k,v 1 ) E. A chord of a cycle [v 1,...,v k,v 1 ] is an edge between two vertices of the cycle that is not an edge of the cycle. A cycle is chordless if it contains no chords. Trivially, a triangle has no chord, so we refer to a chordless cycle in this work as having length strictly greater than 3. We denote by C k the chordless cycle on k vertices. Definition An undirected graph G is chordal (triangulated) graph, if every cycle in G of length strictly greater than 3 possesses a chord. A graph G = (V,E) is a weakly chordal graph if neither G nor its complement G have an induced subgraph C k, k 5 (see [2], [3]). We often use the notation P H[v,w] to denote a chordless path [v = v 1,...,v k = w] in the induced subgraph H, that is, v 1,...,v k V (H) and {(v 1,v 2 ),...,(v k 1,v k )} = E(H {v1,...,v k }). If C = [v 1,...,v k,v 1 ] is a chordless cycle in G (the labels are assigned according to clockwise order), then we denote by P C[v,w] a clockwise path of the cycle C starting with vertex v C and ending with the vertex w C. In this paper, we introduce the family of chordal probe graphs, a generalization of interval probe graphs, defined as follows. Definition An undirected graph G = (V,E) is a chordal probe graph if its vertex set can be partitioned into two subsets, P(probes) and N(non-probes), where N is a stable set and there exists a completion E {(u,v) u,v N, u v} such that G = (V,E E ) is a chordal graph. Example The graph C 4 is not chordal by definition. However, it is chordal probe as shown in Figure 1, since there exists a partition P = {c,d}, N = {a,b} that can be completed into a chordal graph by adding an edge (a,b). Caesarea Rothschild Institute of Computer Science, University of Haifa, Haifa, Israel golumbic@cs.haifa.ac.il 1

2 a c d b probe non-probe non-probe or probe The dotted edge is the completed edge Figure 1. In general, any bipartite graph is a chordal probe graph, by filling one of the stable sets of the bipartition into a clique. An undirected graph G = (V,E) is an interval graph if its vertices can be put into one-to-one correspondence with a set of intervals I = {I v } v V of a linearly ordered set (like the real line) such that two vertices are adjacent in G if and only if the corresponding intervals have a non-empty intersection that is, (u,v) E I u I v (see[3]). Three vertices of G form an asteroidal triple of G if for every pair of them there is a path connecting these two vertices that avoids the neighborhood of the remaining vertex. Theorem (Lekkerkerker and Boland) A graph G is an interval graph if and only if it is chordal and does not contain an asteroidal triple [3]. Interval probe graphs, a generalization of interval graphs, were introduced by Zhang [11] and used in [12] to model certain problems in physical mapping of DNA when only partial data is available on the overlap of clones (i.e., the intervals)(see [6],[10]). Definition An undirected graph G = (V,E) is a interval probe graph if its vertex set can be partitioned into two subsets, P(probes) and N(non-probes), where N is a stable set and there exists a completion E {(u,v) u,v N, u v} such that G = (V,E E ) is an interval graph. Remark 1 Interval probe graphs are chordal probe graphs. Any interval completion is also a chordal graph completion. The converse is not true, as follows. The even length chordless cycles greater than 4 are chordal probe but not interval probe graphs, since interval probe graphs are weakly chordal (see [6],[10]). The odd chordless cycles of length > 4, however, are not chordal probe graphs as shown in Section 2. The definitions of interval probe and chordal probe graphs do not specify a particular partition of the vertices in advance. However, in the biology applications, the partition into probes and nonprobes is part of the input. Hence, we may distinguish between the general case of interval probe or chordal probe graphs, where we must find both a partition and a completion for it, and the special case of partitioned interval probe or partitioned chordal probe graphs, where we are given a fixed partition and must only find a completion for it. A polynomial time algorithm for the problem of recognizing partitioned interval probe graphs (i.e., with respect to a fixed partition) was first reported in [8]. Their method uses PQtrees and constructs an interval probe model in O( V 2 ) time. Another method given in [9], uses modular decomposition and has complexity O( V + E log V ). In contrast to this, however, the complexity of the general problem of recognizing interval probe graphs (when no partition is given) is an open problem. In this paper, we give O(m 2 ) algorithms for recognizing whether a graph is a weakly chordal, chordal probe graph, first in the partitioned case (Section 3.1) and second in the general case (Section 3.3). Definition The graph G = (V,E) is a tolerance graph if each vertex v V can be assigned a closed interval I v and a tolerance t v R + so that (x,y) E if and only if I x I y min(t x,t y ), see [6]. Every interval probe graph is a tolerance graph, by assigning infinite tolerances to non-probes and very small tolerances to probes. The complexity of recognizing tolerance graphs is an open problem. In [4], we presented the hierarchy of tolerance, interval probe and interval graphs, and the restricted cases of having an interval representation where (i) intervals have unit length or (ii) no interval properly contains another interval, see also [6]. In Section 2, we give the complete hierarchy for the classes of chordal probe, weakly chordal, interval probe and related families of graphs. 2

3 2 Structural Results Lemma 2 If G is a chordal probe graph with respect to a given partition (P,N), where N is a stable set, then probes and non-probes alternate in every chordless cycle in G. Proof: Suppose there is a chordless cycle C = [x 1,...,x k,x 1 ] in G (k 4 and where the labels are assigned according to clockwise order), such that probes and non-probes don t alternate in C. Since N is a stable set, there exists a pair {x 1,x 2 } of adjacent probes in C. The probes x 1 and x 2 remain with the same neighborhood in any chordal completion graph G of G. There exists a chordless cycle in G, which consists of [x k,x 1,x 2,x 3 ] together with the chordless path from x 3 to x k. This contradicts the chordality of G. Hence, the given partition (P,N) does not have a chordal completion, contradicting the assumption. Hence, probes and non-probes alternate along the cycle C. Lemma 2 implies the following Theorem. Theorem 3 If G is a chordal probe graph, then G has no induced subgraph C 2k+1, for k 2. Theorem 4 If G is a chordal probe graph, then G has no induced subgraph C k, for k 5. Proof: Let G = (V,E) be a chordal probe graph. By Theorem 3, G has no induced C 5, since C 5 is isomorphic to C 5. Suppose there exists k 6, such that C k = [x 1,...,x k,x 1 ] is an induced subgraph of G (ordered according to clockwise direction). The cycle C = [x 2,x 4,x 1,x 5,x 2 ] is chordless in G. In any partition of V (G) into probes and non-probes, which has chordal completion, either x 1 and x 2 are non-probes or x 4 and x 5 are non-probes. Without loss of generality, assume that x 1 and x 2 are non-probes. The vertex x 1 is adjacent to all the vertices in C k, except for x 2 and x k. Hence, all the vertices in C k, except possibly x 2 and x k are probes. But C = [x 3,x 5,x 2,x 6,x 3 ] is also a chordless cycle of length 4 in G. In any partition (P,N) of V (G), which has chordal completion, either x 2 and x 3 are non-probes or x 5 and x 6 are non-probes. Contradiction. The complete hierarchy of chordal probe graphs and other well-studied families of graphs are summarized in Figure 2. We say that a hierarchy is complete, when all containment relationships are given. That is, (1) a downward edge from class A to class B indicates that class A contains class B, (2) the lack of a hierarchical (containment) relation indicates that the classes are incomparable, (3) classes that appear in the same box are equivalent, (4) an example appearing along the edge between two classes is a separating example for those classes. See [5] for details. Definition A graph G = (V,E) is an even-chordal graph if it has no induced chordless cycle of even length > 4. (This is what would be called (1,6)-even-chordal in [2]). Corollary 5 If G is a chordal probe graph, then G is even-chordal if and only if G is weakly chordal. Definition Let C = [a,c,b,d,a] be a chordless cycle in G, such that probes and non-probes alternate in C, as shown in Figure 1. Any chordal completion of G has an edge (a,b), which we call an enhanced edge and the probes {c,d} are called the creator pair of the enhanced edge (a,b). The enhanced graph G = (P N,E ) is the graph G together with all enhanced edges 1. Definition A partition of a vertex set of a graph into two subsets P (probes) and N (non-probes) is valid if N is a stable set and probes and non-probes alternate in every chordless cycle in G. We now present the main result of this section. Theorem 6 Let G = (V,E) be an even-chordal graph. If there exists a valid partition of V (G), then G is a chordal probe graph and the enhanced graph G is a chordal completion. Proof: Let G = (P,N,E) be an even-chordal graph with a given valid partition of V (G) into P and N. By the definition of valid partition, there are no chordless cycles of odd length in G, hence all chordless cycles in G are 4-cycles. We will show by contradiction that G has no induced chordless cycle. Suppose C is a chordless cycle in G, with the smallest possible number t of enhanced edges. 1 The notion of the enhanced graph was first introduced for interval probe graphs in [11], (see [10]). We apply it more generally. 3

4 If C has no enhanced edge (t = 0), then C is a 4-cycle in G. Since probes and non-probes alternate in C, it must have enhanced edge. Contradiction. Thus, t 1 and there is no chordless cycle in G with less than t enhanced edges. Let {(a 1,b 1 ),...,(a t,b t )} be the enhanced edges in C. We prove the two following claims needed to complete the proof. C6 C 7 H1 Even-chordal Chordal probe C7 Weakly chordal H1 C6 C3 T3 Tolerance Chordal probe Even-chordal Chordal probe Weakly-chordal H1 H2 Interval probe C 4 C4 Chordal Bipartite T 3 Interval C6 C3 a b a d c e c b d f e f g H1 H2 T3 Figure2. The complete hierarchy between chordal probe and other well-studied families of graphs. Claim 7 Let {c,d} be a creator pair of an enhanced edge (a,b) in C. At most one of c or d may have a neighbor x C, such that x a, x b. Proof of Claim 7: Suppose both c and d have a common neighbor x C, such that x a and x b, as shown in Figure 3a. The vertex x cannot be adjacent to both a and b, because C is a chordless cycle with at least 4 vertices. If (a,x) E, then let C = [c,a,d,x,c], else let C = [c,b,d,x,c]. C is a chordless cycle of length 4 in G and also in G. Hence, probes and non-probes alternate in C, and consequently it must have an enhanced edge. Contradiction! Suppose c and d have no common neighbor, then let P = [a = x 1,...,x t = b] be the longer path from a to b on the cycle C. Let x i be a first vertex on P, which is a neighbor of either c or d, and let x j be the next on P which is a neighbor of the other of d or c, as shown in the Figure 3b. Then C = [a,...,x i,...,x j,d,a] is a chordless cycle in G, with less than t enhanced edges. Contradiction. This proves Claim 7. Claim 8 No creator pair creates more than one enhanced edge in C. In other words there are no two equal creator pairs {c i,d i } = {c j,d j }, i j, which create (a i,b i ) and (a j,b j ) respectively. Proof of Claim 8: Suppose there exist creators {c i,d i } = {c j,d j }, i j, which create (a i,b i ) and (a j,b j ) respectively, as shown in Figure 3c, while possibly a i = a j or b i = b j. Then the pair c i,d i also creates the enhanced edges (a i,b j ), (b i,a j ), and (b i,b j ) if b i b j and (a i,a j ) if a i a j. This contradicts the assumption that C is a chordless cycle in G. This proves Claim 8. 4

5 c We now complete the proof of the theorem. By Claim 7, there exists a creator d i for every enhanced edge (a i,b i ), such that d i is adjacent only to vertices a i and b i in C. All the elements of X = {d 1,...,d t } are different because their neighborhoods are different. If X is an independent set of G, then by replacing each enhanced edge (a i,b i ) by the two edges (a i,d i ),(d i,b i ) E(G), we would obtain a chordless cycle C of G of length > 4. Contradiction! Otherwise, (by renumbering if necessary) there exist probes d 1 and d i,i > 1, such that (d 1,d i ) E(G). If a 1 = b i, then b 1 a i due to Claim 8. If a 1 = b i, then assign C = [(b 1,d 1 ),(d 1,d i ),(d i,a i ),E(P C[ai,b 1])], else assign C = [(a 1,d 1 ),(d 1,d i ),(d i,b i ),E(P C[bi,a 1])]. The cycle C is a chordless cycle in G with number of enhanced edges less than t. Contradiction! Thus, we have shown that the enhanced graph G of G with respect to the given valid partition is a chordal graph. Therefore, it is a chordal completion and G is a chordal probe graph. c d c d ci = c j a b a b a i bi a j b j x (a) xi xj (b) Figure 3. d i = d j (c) Corollary 9 Let G = (P,N,E) be a chordal probe graph with respect to a given partition (P,N), where N is a stable set. If G is an even-chordal graph, then the enhanced graph G is chordal. Proof: According to Lemma 2, probes and non-probes alternate in every cycle greater than 3 in G. Since G is an even-chordal graph, then according to Theorem 6 the enhanced graph G is chordal. By Remark 1, every interval probe graph is a chordal probe graph with respect to the same partition and is also an even-chordal graph. Therefore, we obtain an alternate proof of the following: Corollary 10 (Zhang [11]) Let G = (V, E) be an interval probe graph with respect to the partition of its vertex set into P and N, where N is a stable set. The enhanced graph G is chordal. 3 Algorithmic aspects This section deals with recognition of those graphs which are chordal probe graphs and also even-chordal graphs. As shown in Section 2, this class properly contains the class of interval probe graphs. We consider both the case with respect to a given partition (section 3.3) and the more challenging case without being given the partition (section 3.1). Remark 11 A chordless cycle of length 4 (i.e., 4-cycle) has exactly two valid partitions where either pair of non-adjacent vertices could be the non-probes. Moreover, assigning any one of its four vertices to be a probe (respectively non-probe) forces its neighbors in the cycle to be non-probes (resp. probes) and its non-neighbor to be a probe (resp. non-probe). Remark 12 The number of 4-cycles in a graph is at most O( E(G) 2 ) and generating them can be done in O( E(G) 2 ) time. 3.1 Recognition of those chordal probe graphs which are also even-chordal with respect to a given partition According to Corollary 5, a chordal probe graph is weakly chordal if and only if it is evenchordal. The recognition of weakly chordal graph can be done in O( E(G) 2 ) ([1] or [7]). We use this method at the first stage of our algorithm. At the second stage, for each 4-cycle, we verify that probes and non-probes alternate on this cycle. The algorithm may fail at first stage, meaning that the graph is not weakly chordal. The algorithm may fail at the second stage, then the graph is weakly chordal, but is not chordal probe, since probes and non-probes do not alternate in every chordless cycle in G. The overall time complexity of the algorithm is O( E(G) 2 ) by Remark 12. 5

6 3.2 The C 4 -connectivity relation Definition Let G = (V,E) be a connected graph and let S(G) denote the set of all 4-cycles in G. We define the sets S x (G) = {C S(G) x V (C)} for each x V (G). A path [v 1,...,v i,...,v n ] in G is a C 4 -path if there exists C i S(G) such that (v i,v i+1 ) E(C i ) for each i = 1,...,n 1. A pair of vertices is C 4 -connected if there exists a C 4 -path that connects the vertices. A graph is C 4 -connected graph if each pair of its vertices is C 4 -connected. The C 4 -connectivity is an equivalence relation on V, so it partitions the set V into vertex disjoint maximal C 4 -connected components, which we call C 4 -components. A C 4 -component which has only one vertex is called a singleton C 4 -component. Let H 1,...,H t be the C 4 -components of G. The edges {(x,y) x H i and y H i, i = 1,...,t} are called the internal edges of the graph. For each x H i we define the set N j (x) = {y H j (x,y) E(G), j i} and each such edge (x,y) is called an external edge. If N j (x) = {y} and N i (y) = {x}, then (x,y) is an exclusive external edge and the vertices x and y are called exclusive endpoints. See Figure 4. Lemma 13 If (P 1,N 1 ),(P 2,N 2 ) are two different partitions of a C 4 -connected graph G, such that probes and non-probes alternate in every chordless cycle in G, then P 1 = N 2 and N 1 = P 2. Proof: Suppose there exists v V (G), which is either assigned to be a probe in both partitions or a non-probe in both partitions. We will now prove by induction on the length of a C 4 -path P = [v = v 0,...,v i,...,v m ] that the vertex v m has the same assignment in both partitions. In the case that P = [v 0,v 1 ], the edge (v 0,v 1 ) is an edge set of a 4-cycle. Therefore, v 1 has the same assignment in both partitions due to Remark 11. Assume that for every path of length i < m, the vertex v i has the same assignment in both partitions. Let P = [v = v 0,...,v m ] be a C 4 -path of length m. By induction, each vertex v i (1 i < m) must have the same assignment in both partitions. Consequently, the vertex v m must have the same assignment in both partitions, since the length of the c 4 -path [v i,...,v m ] is less than m. Therefore, all the vertices in G have the same assignment in both partitions and hence P 1 = P 2 and N 1 = N 2. Contradiction! Thus, the assignment of each vertex in (P 1,N 1 ) is different than its assignment in (P 2,N 2 ), so P 1 = N 2 and P 2 = N 1. Lemma 14 Let G be a weakly chordal graph, and H i be a C 4 -component of G. Then the induced subgraph G Nj(x) is connected for any x H i (i j). Lemma 15 There is at most one exclusive external edge that connects two C 4 -components in G. Proof: Suppose (u,v) and (a,b) are exclusive external edges that connect H i and H j, where u,a H i and v,b H j, by definition u a and v b. Let x be the first vertex on P Hi[a,u], which has neighbors on P Hj[b,v], and let y be the first vertex on P Hj[b,v] which is adjacent to the vertex x (y b since (a,b) is an exclusive external edge). The cycle C = [a,p Hi[a,x],x,y,P Hj[y,b],b,a] is chordless of length > 3 and has vertices from different C 4 -components. Contradiction. We use the FindC4Components procedure to find all the C 4 -components in a graph. The procedure is a variant of breadth first search, and in O( E(G) 2 ) time it combines those 4-cycles that are not vertex disjoint into a C 4 -component. Those vertices which are not in a vertex set of a 4-cycle are singleton C 4 -components. See [5] for more details. Example H 1,...,H 8 are the C 4 -components in the graph shown in Figure 4, where H 1, H 2, H 3, H 6 are singleton C 4 -components. The C 4 -component H 5 has only one valid partition. H1 H4 H5 H6 H7 H8 non-exclusive external edge exclusive external edge internal edge H2 H3 Figure 4. Example 6

7 3.3 Recognition of those chordal probe graphs which are also even-chordal without being given the partition In the first stage of our algorithm, we test whether G is a weakly chordal graph, using the method in [1] or [7]. In the second stage, for each vertex x in the graph, we find the set S x (G) of 4-cycles containing x, and in the third stage, we construct the C 4 -components H 1,...,H t of G. Stage four finds all valid partitions of each H i. If there exists a C 4 -component H i, which does not have a valid partition, then H i is not a chordal probe graph due to Lemma 2 and hence G is not a chordal probe graph by the hereditary property. At the fifth stage, we extend the partitions from the C 4 -components to the entire graph, and upon success we construct a graph G 1 = (V 1,E 1 ), whose vertices correspond to the vertices of G and edges correspond to the internal edges and a certain subset of exclusive external edges of G. We will prove that if stages 1-5 succeed, then G 1 is a chordal probe graph. Finally, we will show that G 1 is a chordal probe graph if and only if G is a chordal probe graph. Algorithm II. Recognition of chordal probe graphs which are even-chordal graphs without being given the partition 1. verify that G = (V, E) is weakly chordal using the algorithm in [1] or [7], otherwise return failure ; 2. construct the set S(G) of all 4-cycles in G and the sets S x(g) for each x V (G); 3. find the set H = H 1,..., H t of C 4-components of G by calling the FindC4Components procedure; 4. for each H i H do if H i is a singleton C 4-component then P i = V (H i), N i =, l i = 1; else find a valid partition (P i, N i) and the number l i of valid partitions of H i by calling the FindPartitions procedure; if it fails, then return failure ; 5. build the graph G 1 = (V 1, E 1) by calling the PropagateConstrainedGraph procedure. If it fails, then return failure, otherwise return success ; We now give the details of the procedures that are used in the algorithm. The FindPartitions procedure finds all the valid partitions of a weakly chordal C 4 -connected component H i and assigns a label l i, namely 1 or 2. If H i does not have a valid partition, then the FindPartitions procedure fails. The procedure first finds a partition of vertices, such that probes and non-probes alternate on every chordless cycle. It makes an arbitrary vertex v to be a probe and then propagates the assignment of all the other vertices accordingly. Applying Lemma 13, there exist at most two such partitions of H i, where in the case of two valid partitions the set of probes in one of the partitions is exactly the set of non-probes in the other partition. Then the FindPartitions procedure checks if non-probes are a stable set and if probes are a stable set in the partition. Procedure: FindPartitions Input: A C 4-component H i and the set S x(g) for each x V (G). Output: A valid partition (P i, N i) of H i and the label l i, or failure. step a (P i, N i) ({v}, ), where v is an arbitrary vertex in H i; l i 0; insert the vertex v into a queue Q; while the queue Q is not empty do remove vertex u from Q; for each cycle C i S u(g) do if the non-neighbor x of u in C i has a different assignment than u, then return failure ; if x is not yet assigned then assign x to have the same assignment as u and insert x into Q; 7

8 step b if either neighbor y 1 or y 2 of u in C i has the same assignment as u, then return failure ; if either y i is not yet assigned then assign y i to have different assignment than u and insert y i into Q; if N i is a stable set then if P i is a stable set then l i 2, else l i 1; else if P i is a stable set, then swap the sets P i and N i and l i 1; else return failure ; Lemma 16 Let H i be a non-singleton C 4 -component of a weakly chordal graph G. The FindPartitions procedure finds all the valid partitions of H i. Proof: Each partition that is found by the FindPartitions procedure is valid, since probes and non-probes alternate in every 4-cycle (this is checked in step (a)) and non-probes are a stable set (this is checked in step (b)). According to Lemma 13, there exists at most two valid partitions, while the set of probes in one of the partitions is exactly the set of non-probes in the other partition. The FindPartitions procedure checks if both partitions are valid, by checking if the probes and the non-probes are stable sets. The PropagateConstrainedGraph procedure identifies the external edges of G and builds the graph G 1 = (V 1,E 1 ), enforcing the constraint that the set of non-probes must be a stable set. In the process, a vertex can be forced to be a probe using the ForceVertexToBeAProbe procedure, which assigns a valid partition to the C 4 -component, such that the vertex is a probe and reduces its label to 1, or fails. Procedure: PropagateConstrainedGraph Input: The set H = H 1,..., H t of C 4-components in a graph G = (V, E), and a valid partition {P i, N i} of H i together with the label l i, for all i. Output: Finds the graph G 1 = (V 1, E 1), or failure /* Step I: marking of internal edges */ for each H i, mark the internal edges E(H i); /* Step II: marking of non-exclusive external edges */ for each v V (G), let v H i do for each unmarked edge (u, v) E(G), u N j(v), N j(v) > 1 do mark the edge (u, v); if there is at least one non-probe in N j(v) then call the ForceVertexToBeAProbe(v) procedure; /* Step III: marking of some exclusive external edges */ insert all C 4-components with label equal to 1 into the queue Q; while Q is not empty do remove component H i from Q; for each v H i do if there exists an unmarked edge (u, v) E(G), u N j(v) then mark the edge (u, v); if u,v are both non-probes then call the ForceVertexToBeAProbe(u); insert H j into Q; let G 1 = (V 1, E 1) be a graph whose vertices correspond to vertices of G (V 1 = V ) and edges correspond to the remaining unmarked edges of G together with all internal edges E(H i). We omit the proves of the following properties, which can be found in [5]. 8

9 Lemma 17 If the PropagateConstrainedGraph procedure succeeds, and e is an external edge of G, then e E 1 if and only if it is an exclusive external edge that connects two C 4 -components both having two valid partitions remaining at the end of the procedure. Lemma 18 Every chordless cycle in G 1 is an induced subgraph of a C 4 -component of G. Definition A vertex is called simplicial if its adjacency set is a clique. Let G = (V,E) be an undirected graph and let σ = [v 1,...,v n ] be an ordering of the vertices. We say that σ is a perfect elimination ordering(peo) if each v i is a simplicial vertex of the induced subgraph G {vi,...,v n}. Theorem (Fulkerson and Gross) A graph is chordal if and only if it has a PEO [3]. Let G 2 = (V 2,E 2 ) be the quotient graph of G 1, where each h i V 2 corresponds to the C 4 -component H i in G and E 2 = {(h i,h j ) (u,v) E 1 such that u H i,v H j }. Remark 19 Obviously G 2 is a chordal graph, since every chordless cycle in G 1 is contained in a C 4 -component due to Lemma 18. Therefore, there exists a PEO σ = [h 1,...,h t ] of V 2, which corresponds to an ordering ν = [H 1,...,H t ] of the C 4 -components in G. Lemma 20 Let G be a weakly chordal graph and ν = [H 1,...,H t ] the ordering of the C 4 -components in G, which corresponds to a PEO in G 2. There exists at most one exclusive endpoint of an edge in H i that connects H i with any of the C 4 -components [H i+1,...,h t ] in G 1. Proof: Suppose there exists exclusive external edges (x 1,y 1 ) and (x 2,y 2 ), x 1 x 2, such that x 1,x 2 H i, y 1 H j, y 2 H k and j,k > i. Then j k according to Lemma 15. Since h i is a simplicial vertex in G 2[hi+1,...,h t], there exists an exclusive external edge (z 1,z 2 ), z 1 H j, z 2 H k, as demonstrated in Figure 5 (possibly y 1 = z 1 or y 2 = z 2 ). The chordless cycle C = [x 1,P Hi[x 1,x 2], x 2,y 2,P Hk [y 2,z 2],z 2,z 1,P Hj[z 1,y 1],y 1, x 1 ] in G has vertices in different C 4 -components. Contradiction. Theorem 21 The PropagateConstrainedGraph procedure succeeds if and only if G is a chordal probe graph. Proof: ( ) If the PropagateConstrainedGraph procedure succeeds, then by Remark 19, let σ be PEO of G 2 that corresponds to an ordering ν of the C 4 -components in G. If h i is an isolated vertex in G 2, then (P i,n i ) is a valid partition of H i. Otherwise, let x be the exclusive endpoint in H i that connects H i to H j, for all j > i in G 1, as in Lemma 20. There exist two valid partitions of H i due to Lemma 17. Let (P i,n i ) be the valid partition of H i, such that x P i. Consider the partition (P = (P i ),N = (N i )) of V (G 1 ). Probes and non-probes alternate in every cycle of length > 3 in G, since such a cycle is an induced subgraph in a C 4 -component due to Lemma 18. Suppose there exists an edge (u,v) E 1, such that u N i, v N j and j > i. Then u is the exclusive endpoint of H i and hence u P j, a contradiction. Thus N is a stable set and (P,N) is a valid partition of G 1. Therefore, G 1 is a chordal probe graph due to Theorem 6. Let (P = (P i ),N = (N i )) be a valid partition of G 1, where (P i,n i ) is a valid partition of H i. We will prove that (P,N) is a valid partition of G and hence G is a chordal probe graph due to Theorem 6. Since G is a weakly chordal graph, each chordless cycle of length > 3 in G is an induced subgraph of a C 4 -component and hence probes and non-probes alternate in the cycle. Thus, we only need to prove that N is a stable set. Suppose for a contradiction that there exists an edge (u,v) E(G), such that u,v N. Moreover, u H i and v H j, since N i is a stable set for all i. In case that N j (v) > 1, the procedure either fails or forces the vertex u to be a probe. Therefore N j (v) = 1 and similarly N i (v) = 1. Thus, (u,v) is an exclusive external edge. Neither H i nor H j were inserted into Q, since otherwise the procedure would remove H i (or H j ) from Q and force v (or u) to be a probe. Thus (u,v) is unmarked by the procedure, meaning that (u,v) E 1 and N is not a stable set in G 1. This is a contradiction, since G 1 is a chordal probe graph with respect to (P,N). ( ) We prove that if the procedure fails, then G is not a chordal probe graph. In the case that the PropagateConstrainedGraph procedure fails at Step II, there exists an edge (u,v) E(G), v H i, u N j (v), N j (v) > 1, such that there is at least one non-probe in N j (v) in a given partition (P j,n j ), v is a non-probe in a given partition (P i,n i ) and l i = 1. Since N j (v) is a connected set by Lemma 14 and N j is a stable set, u has a probe neighbor in N j (v) in the given partition (P j,n j ). Thus, there is also a non-probe in N j (v) in the opposite partition of H j. Therefore, there exists a 9

10 non-probe in N j (v) in any valid partition of H j. Suppose G is a chordal probe graph. Now, (P i,n i ) is the only valid partition of H i, since v must be a non-probe in any valid partition of G. Thus there would be a pair of adjacent non-probes in any valid partition of G, a contradiction. In the case that the PropagateConstrainedGraph procedure fails at Step III, there exists an edge (u,v) E(G), u N j (v), N j (v) = 1, where u,v are both non-probes in the given partitions (P i,n i ), (P j,n j ) and l j = l i = 1. Suppose G is a chordal probe graph. Since both H i and H j have only one valid partition, u and v are both non-probes in any valid partition of G. Thus there would be a pair of adjacent non-probes in any valid partition of G, a contradiction. Remark 22 The time complexity of the Algorithm II is O( E(G) 2 ). Proof: The first stage of the algorithm has time complexity O( E(G) 2 ), using the algorithm in [1] or [7]. According to Remark 12, the second stage also has time complexity O( E(G) 2 ). At the third stage, we call the FindC4Components procedure, which has complexity O( E(G) 2 ). At the fourth stage, the FindPartitions procedure is called for each C 4 -component H i of the graph. The time complexity of FindPartitions procedure is O( E(H i ) 2 ) (see [5]), and hence the fourth stage has time complexity i O( E(H i) 2 ) = O( E(G) 2 ). The time complexity of the fifth stage is O( E(G) ) (see [5] for details). Hi x1 y 1 z 1 H j x 2 z 2 y 2 H k Figure 5. References [1] A. Berry, J. Bordat, P. Heggernes, Recognizing weakly triangulated graphs by edge separability, Nordic Journal of Computing 7(3), Fall(2000), [2] A. Brändstadt, V. Le, J.P. Spinrad, Graph classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, [3] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York(1980). [4] M.C. Golumbic, M. Lipshteyn, On the hierarchy of tolerance, probe and interval graphs, Congressus Numerantium 153 (2001), [5] M.C. Golumbic, M. Lipshteyn, Chordal probe graphs, manuscript (2003). [6] M.C. Golumbic, A.N. Trenk, Tolerance Graphs, Cambridge University Press (2003). [7] R.B. Hayward, J. Spinrad, R. Sritharan, Weakly chordal graph algorithms via handles, Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms (2000), [8] J.L. Johnson, J. Spinrad, A polynomial time recognition algorithm for probe interval graphs, Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (2001), [9] R.M. McConnell, J. Spinrad, Construction of probe interval models, Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms (2002), [10] F.R. McMorris, C. Wang, P. Zhang, On probe interval graphs, Discrete Applied Mathematics 88 (1998), [11] P. Zhang, Probe interval graph and its application to physical mapping of DNA, manuscript [12] P. Zhang, E.A. Schon, S.G. Fischer, E. Cayanis, J. Weiss, S. Kistler, and P.E. Bourne, An algorithm based on graph theory for the assembly of contigs in physical mapping of DNA., CABIOS 10 (1994),