Discrete Applied Mathematics. The induced path function, monotonicity and betweenness

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1 Discrete Applied Mathematics 158 (2010) Contents lists available at ScienceDirect Discrete Applied Mathematics jornal homepage: The indced path fnction, monotonicity and betweenness Manoj Changat a, Joseph Mathew b, Henry Martyn Mlder c, a Department of Ftres Stdies, University of Kerala, Trivandrm , India b Department of Mathematics, S.B. College, Changanassery , India c Econometrisch Institt, Erasms Universiteit, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands a r t i c l e i n f o a b s t r a c t Article history: Received 21 Jne 2006 Received in revised form 4 October 2009 Accepted 17 October 2009 Available online 12 November 2009 Keywords: Transit fnction Indced path Betweenness Monotone Long cycle Hose Domino P-graph The geodesic interval fnction I of a connected graph allows an axiomatic characterization involving axioms on the fnction only, withot any reference to distance, as was shown by Nebeský [20]. Srprisingly, Nebeský [23] showed that, if no frther restrictions are imposed, the indced path fnction J of a connected graph G does not allow sch an axiomatic characterization. Here J(, v) consists of the set of vertices lying on the indced paths between and v. This fnction is a special instance of a transit fnction. In this paper we address the qestion what kind of restrictions cold be imposed to obtain axiomatic characterizations of J. The fnction J satisfies betweenness if w J(, v), with w, implies J(w, v) and x J(, v) implies J(, x) J(, v). It is monotone if x, y J(, v) implies J(x, y) J(, v). In the case where we restrict orselves to fnctions J that satisfy betweenness, or monotonicity, we are able to provide sch axiomatic characterizations of J by transit axioms only. The graphs involved can all be characterized by forbidden sbgraphs Elsevier B.V. All rights reserved. 1. Introdction In [18] the notion of transit fnction is introdced as a means to stdy how to move arond in discrete strctres. Basically, it is a fnction satisfying three simple axioms on a set V, which is provided with a strctre σ. Prime examples of sch strctres are: a set of edges E, so that we are considering a graph G = (V, E), or a partial ordering, so that we are considering a partially ordered set (V, ). The idea is to stdy transit fnctions that have additional properties defined in terms of the strctre σ. For instance, the transit fnction may be defined in terms of paths in the graph G = (V, E). Sch transit fnctions are called path transit fnctions on G in [18]. A prime example is the geodesic interval fnction I : V V 2 V of a connected graph G, where I(, v) is the set of vertices lying on the shortest paths between and v. This fnction has been widely stdied from many different perspectives, to name a few: convexity, see e.g. [10,17,29], medians, see e.g. [14,17], monotonicity, see e.g. [15,17,24]. For the indced path fnction J : V V 2 V of a connected graph G, where J(, v) is the set of vertices lying on the indced paths between and v, similar qestions and problems have been stdied: convexity, see e.g. [4,9,11,13,16], median-type properties, see [16], monotonicity, see e.g. [3 5]. This exemplifies the basic idea for introdcing the concept of transit fnction in [18]: transfer ideas, qestions and problems from one transit fnction to another and see whether interesting problems arise. This was the motivation to stdy the analoges of these qestions for the all-paths fnction A on a graph: now A(, v) consists of the vertices on the, v-paths, see [2]. The convexity related to the all-paths fnction was already stdied mch earlier, see e.g. [8,26]. Note that any transit fnction has an associated convexity. Sch convexities are called interval convexities in [1,29]. Those related to path transit fnctions are discssed in more detail in [6]. Corresponding athor. addresses: mchangat@gmail.com (M. Changat), hmmlder@few.er.nl (H.M. Mlder) X/$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.dam

2 M. Changat et al. / Discrete Applied Mathematics 158 (2010) v y w x A B C Fig. 1. A: hose, B: domino, C: P-graph. In [20 22] Nebeský obtained some qite interesting reslts, see also [19]. He characterized the fnctions that are the geodesic interval fnction of some graph withot any reference to the notion of distance. That is, a fnction I : V V 2 V is the geodesic interval fnction of some connected graph if and only if I satisfies a set of axioms that are phrased in terms of I only. This immediately poses the problem for other transit fnctions on graphs: can they be characterized in terms of sch transit axioms only? For the all-paths fnction A this was done in [2]. Srprisingly, sch a characterization of the indced path fnction J is not possible, as was shown by Nebeský in [23] sing first order logic. This poses the problem whether it is still possible to characterize the indced path fnction if some frther restrictions are imposed, or if the graph satisfies some extra properties. The aim of this paper is to stdy special cases, in which J can indeed be characterized by transit axioms only. Then one searches for the appropriate properties of the graphs and the appropriate transit axioms for J. These cases are where J has the properties of a betweenness, and where J is monotone, that is, all sets J(, v) are J-convex. As one might expect, the characterizations we seek for J in this paper involve forbidden (indced) sbgraphs for the graphs. The most important ones are the hose, the domino and the P-graph, see Fig. 1, and the holes. Here a hole, or a long cycle, is a cycle with at least 5 vertices. The so-called HHD-free graphs and HHP-free graphs that appear over and over below also have other interesting aspects. Here H stands for hose or hole, D for domino, and P for P-graph. These classes of graphs have important applications as far as elimination orderings in graphs are concerned. HHD-free and HHP-free graphs are natral generalizations of the class of chordal graphs in connection with the lexicographic breadth first search (LexBFS) and maximm cardinality search (MCS) orderings in graphs, see [25,28]. In [7], sing a relaxation of the indced path convexity known as m 3 -convexity, it is proved that graphs, for which LexBFS (MCS) is a semi-simplicial ordering, constitte precisely the class of HHD-free (HHP-free) graphs. See also [12]. The paper is organized as follows. In Section 2 we give the definition of transit fnction, betweenness and monotonicity, and introdce five new axioms for the characterization of the indced path fnction J in terms of these transit axioms. Each of these new axioms captres some aspect of the idea of betweenness that is exemplified in the geodesic interval fnction. Moreover we prove some first reslts involving J and betweenness and monotonicity. In Section 4 we prove or main reslts, viz. Theorems 2 and 3: a transit fnction that is a betweenness and satisfies in addition some of the five new axioms necessarily is the indced path fnction of some connected graph. Using the above characterizations, we also characterize the classes of HHD-free and HHP-free graphs by the indced path fnction. 2. Transit fnctions and betweenness In this section we collect the necessary terminology on transit fnctions and betweenness and establish some first reslts. A graph is said to be HHD-free if it does not contain a hose, a hole or a domino as an indced sbgraph. It is called HHP-free if it does not contain a hose, a hole or a P-graph as indced sbgraph. A hole is a cycle of length at least 5, for the other graphs see Fig. 1. Let V be a finite set. A transit fnction on V is a fnction R : V V : 2 V satisfying the following three axioms: (t1) R(, v), for any and v in V, (t2) R(, v) = R(v, ), for all and v in V, (t3) R(, ) = {}, for all in V. A sbset W of V is R-convex if R(, v) W, for any two vertices, v in W. If, moreover, G = (V, E) is a graph with vertex set V, then we say that R is a transit fnction on G. Note that the above axioms do not reflect any aspect of the graph G. Bt or interest will be in transit fnctions that are defined in terms of the graph. Then the challenge is whether these graphical properties of the transit fnction can be characterized by transit axioms that are in terms of the transit fnction only. The nderlying graph G R of a transit fnction R is the graph with vertex set V, where two distinct vertices and v are joined by an edge if and only if R(, v) = {, v}. Note that, in general, G and G R will not be isomorphic graphs. Transit

3 428 M. Changat et al. / Discrete Applied Mathematics 158 (2010) fnctions were introdced in [18] to provide a nifying approach to the varios fnctions of this type on graphs stdied in the literatre. Prime examples of transit fnctions on a graph G are the (geodesic) interval fnction I, where I(, v) is the set of vertices on the shortest, v-paths, the all-paths fnction A, where A(, v) is the set of vertices on the, v-paths, and the indced path fnction J, which is defined by J(, v) = {w V w lies on some indced, v-path in G}. Or concern in this paper is with the latter. These three fnctions are the so-called path transit fnctions becase they are defined in terms of paths of G, see [6] and [18] for more information on path transit fnctions. The geodesic intervals I(, v) in G also have the strctre of a betweenness, bt the other two do not. Hence the following betweenness axioms were introdced in [18] to model the idea of betweenness. The first tells s that, if x is between and v bt distinct from v, then v is not between and x. The second tells s that, if x is between and v and y is between and x, then y is between and v. A transit fnction R on V is called a betweenness, if it satisfies (b1) x R(, v), x v v R(, x), (b2) x R(, v) R(, x) R(, v). It is easy to see that the all-paths fnction A is a betweenness on G if and only if G is a tree, see [2]. In [16] it was shown that J is a betweenness on G if and only if G is HHD-free. Note that only few aspects of the betweenness properties of I are reflected in these two axioms. To captre all aspects wold reqire a long and complicated list of axioms. Moreover, we wold not get anything that cold be transferred to other transit fnctions, the whole idea behind the concept of transit fnctions. Therefore, this notion of betweenness is weaker than existing ones in the literatre, see e.g. [27]. If R is a betweenness on V, then we have the following basic lemma. Lemma 1. If the transit fnction R on a non-empty set V is a betweenness, then the nderlying graph G R of R is connected. Proof. Let, v be any two distinct vertices of G R. We prove the existence of a, v-path in G R sing indction on R(, v). If R(, v) = 2, then R(, v) = {, v}, by transit axiom (t1). Therefore, by the definition of G R, we have v E(G R ), which constittes a, v-path in G R. So the lemma holds for R(, v) = 2. Assme that there is a, v-path in G R for any two distinct vertices, v with R(, v) < n (n > 2). Since n > 2, there is a vertex w, v with w R(, v). Hence by (b1) we have R(w, v) and v R(, w). Also by (b2) we have R(, w) R(, v) and R(w, v) R(, v). Therefore R(, w) < R(, v) and R(w, v) < R(, v). Hence, by the indction hypothesis, the existence of a, w-path and a w, v-path follows. Concatenating the two paths we obtain a, v-walk that contains a, v-path. Note that the two betweenness axioms (b1), (b2) are necessary for the connectedness of G R. For example, on V = {a, b, c, d}, the fnction R, defined by R(, ) = {} for every V, R(a, b) = {a, b, c}, R(a, c) = {a, c, d}, R(a, d) = {a, b, d}, R(b, c) = {b, c}, R(b, d) = {b, d}, R(c, d) = {c, d}, is a transit fnction satisfying (b1), bt not (b2) and it can be easily verified that G R is disconnected. On V = {a, b, c}, the fnction R defined by R(a, b) = R(b, c) = R(c, a) = V and R(, ) = {} for every V is a transit fnction satisfying (b2), bt not (b1). Here also G R is disconnected. In Lemma 1 only the connectivity of the nderlying graph is established, bt nothing pertinent can be said yet abot the qestion whether G and G R are isomorphic or not. Moreover, a betweenness in general will not be the indced path fnction of some graph. Hence, we need some more transit axioms for or prposes: the focs on the indced path fnction. An axiom that plays an important role in the stdy of median graphs and median strctres is that of monotonicity, see [17]. There it was introdced for the interval fnction I only, bt in [18] it is introdced as a transit axiom: (m) x, y R(, v) R(x, y) R(, v). Note that in the terminology of convexity this axiom can be read as follows: the R-intervals R(, v) are R-convex. For references on convexity, and monotonicity of I, J, and A, see the Introdction. Now we introdce five new transit axioms, which all reflect some aspect that the betweenness of the fnction I possesses. All five axioms will be sed in the next section. In the rest of this section we focs on axiom (J1) only. Let R be a transit fnction on a connected graph G = (V, E), and let, v, x, y be vertices in V. (J1) w R(, v), w, v, there exists 1 R(, w) \ R(v, w), v 1 R(v, w) \ R(, w), sch that R( 1, w) = { 1, w}, R(v 1, w) = {v 1, w} and w R( 1, v 1 ). (J2) R(, x) = {, x}, R(x, v) = {x, v}, v, R(, v) {, v} x R(, v). (J3) x R(, y), y R(x, v), x y, v, R(, v) {, v} x R(, v). (J2 ) x R(, y), y R(x, v), x y, R(, x) = R(x, y) = R(y, v) = 2, v, R(, v) {, v} x R(, v). (J3 ) x R(, y), y R(x, v), R(x, y) {x, y}, x y, v, R(, v) {, v} x R(, v). Note that, althogh we se the letter J to name these axioms, only the axioms (J2) and (J2 ) are satisfied by the indced path fnction of any graph. Axiom (J1) already captres an essential aspect of betweenness. This is shown by the next reslt and its corollary. In the proof we se the following notation. Let P be a path in a graph G, and let x, y be two vertices on P. Then x P y denotes the sbpath of P between x and y, that is, we walk from x to y along P. A chord of a path is an edge joining two non-consective vertices on the path. So indced paths are precisely the chord-less paths.

4 M. Changat et al. / Discrete Applied Mathematics 158 (2010) v v v p z q p q p q A B C Fig. 2. A: K 2,3, B: W 4 e, C: K2,3. Theorem 1. The indced path fnction J on a graph G satisfies (J1) if and only if G is HHD-free. Proof. First assme that G is not HHD-free. Then G contains a hose, a hole or a domino. In each case we can find three vertices, v and w with and w adjacent and v not adjacent to or w sch that w J(, v) and J(, w) = {, w} J(v, w). Hence we cannot find a 1 as reqired by the axiom (J1). So (J1) is not satisfied. Conversely, assme that (J1) is not satisfied. Take any indced, v-path P = P 1 w v 1 P v with w distinct from and v. Since (J1) is not satisfied, we have 1 J(v, w) or v 1 J(, w). Withot loss of generality, we may assme that 1 J(v, w). Then there exists an indced w, v-path Q containing 1. Evidently Q starts with the edge w 1. Let v 2 be the first vertex on Q which is also a vertex on the path w P v. Then v 2 v 1, otherwise wv 1 will act as a chord of Q. Since P is an indced path, 1 v 1 E(G). Hence Q = 1 Q v 2 is an indced 1, v 2 -path of length greater than or eqal to two and P = 1 w v 1... P v 2 is another indced 1, v 2 -path of length at least three. Together they form a cycle of length at least five. To avoid a long cycle, there mst exist chord between an internal vertex of P and Q. Let v 3 be the vertex on P closest to 1 having a chord to Q, and let v 4 be the vertex on Q closest to 1 having a chord to v 3. Then w v 1 P v 3 v 4 Q 1 w is an indced cycle (say) C. Since C cannot be a long cycle we have v 3 = v 1 and v 4 adjacent to 1. Hence C is an indced cycle of length for. Consider the cycle v 1 P v 2 Q v 4 v 1. If it is of length three or for, then together with C we get a hose or a domino. So it is a cycle of length at least five. Again, to avoid a hole, there mst be chords. As above, we choose a chord closest to v 1 and v 4, which yields a 3-cycle or 4-cycle. Bt now this cycle together with C is a hose or a domino. Ths we have a contradiction, which concldes the proof. The above cited theorem in [16], that the indced path fnction J of a connected graph G is a betweenness if and only if G is HHD-free, gives s the following obvios corollary. Corollary 1. Let J be the indced path fnction of a connected graph G. Then J is a betweenness if and only if J satisfies (J1). Note that the eqivalence of (b1), (b2) on the one hand and (J1) on the other hand in this corollary is a special case that only holds for the indced path fnction of a graph. For arbitrary transit fnctions this eqivalence need not hold. The second betweenness axiom (b2) is a special instance of the monotonicity axiom (m). There are many graphs for which the indced path fnction J satisfies (b2) bt not monotonicity. The smallest sch graph is the complete bipartite graph K 2,3. The family of sch graphs can be constrcted easily from K 2,3. The constrction is as follows. Let and v be the vertices of degree three in K 2,3, and let p, q, z be its vertices of degree two. Sbdivide the path z v and make p adjacent to some internal vertices of z v so that at least the neighbors of and v on this sbdivided path are adjacent to p. Let s denote the reslting graph by K2,3. It can be easily verified that the indced path transit fnction J on K2,3 satisfies (b2) bt not monotonicity. The graphs in Fig. 2 depict some of the graphs in this class. There is one sch graph on five vertices: this one can also be obtained from the wheel on 4 vertices by deleting one of its spokes. Therefore we denote this graph by W 4 e. There are 2 n 6 graphs of type K2,3 on n 6. The following Proposition was proved in [5]. It shows that the graphs K 2,3 and K2,3 are the only graphs for which the indced path transit fnction satisfies (b2) bt not monotonicity. Proposition 1 ([5]). Let G be a connected graph which is K 2,3 - and K2,3 -free. Then the indced path transit fnction J on G satisfies b2 if and only if it is monotone. We already know that the indced path transit fnction J is a betweenness if and only if G is HHD-free. Hence, if J is also monotone, then in view of the above Proposition, the only additional forbidden sbgraphs are K 2,3 and W 4 e. Hence, we have the following immediate corollary. Corollary 2. Let G be a connected graph, and let J be the indced path fnction of G. Then J is a monotone betweenness if and only if G is HHD-free and has no K 2,3 or W 4 e as indced sbgraph.

5 430 M. Changat et al. / Discrete Applied Mathematics 158 (2010) Transit fnctions that are the indced path fnctions of a connected graph In [20] Nebeský characterized the transit fnctions that are the geodesic interval fnction of some graph withot any reference to distance, see also [21,22,19]. Omitting the technical details, the reslt reads as follows. Let I : V V 2 V be a transit fnction on a finite set V. Then I is the geodesic interval fnction of some connected graph G = (V, E) if and only if I satisfies seven transit axioms, which are phrased in terms of the fnction only, hence withot any reference to distance. Using mathematical shorthand, one can also phrase this reslt as follows: let I be a transit fnction on V, and let G I be its nderlying graph. Then the geodesic interval fnction I GI of G I coincides with I, i.e. I = I GI, if and only if I satisfies these seven axioms. This very nice reslt was the inspiration for many papers on path transit fnctions. One might wonder whether a similar reslt holds for the indced path fnction. Srprisingly enogh, in [23] Nebeský proved, sing first order logic, that there does not exist sch a characterization: there is no set of transit axioms sch that J satisfies this set if and only if it is the indced path fnction of some connected graph. Otherwise formlated: Let J be a transit fnction, and let G J be its nderlying graph. Then, in general, the indced path fnction J GJ of G J may be qite different from the original transit fnction J, even if J satisfies axioms reflecting properties of the indced path fnction of a graph. Now the qestion arises whether there are sets of axioms on J sch that these force J = J GJ. This will not be possible for arbitrary graphs, bt it is still possible for special classes of graphs. That is the focs of this section. Note that, if we wold start with a connected graph G, and we wold take the indced path fnction J G of G, then by definition we wold have G JG. Let J be a transit fnction on a non-empty finite set V satisfying some or all of the axioms (b1), (b2), (m), (J1), (J2), (J2 ), (J3), and (J3 ). Using this set of axioms we give two characterizations of the indced path fnction J on the nderlying graph G J. For proving or main theorems we need the following lemmas. Note that the tricky part in the proofs is that we do not know yet whether J is the indced path fnction of G J. Lemma 2. Let J be a transit fnction on a non-empty finite set V satisfying the axioms (b1), (J2) and (J3) with nderlying graph G J. Then G J is HHP-free. Proof. First recall that J(, v) = {, v}, for any edge v in G J. Sppose G J contains a hose as an indced sbgraph with vertices shown in Fig. 1(A). Then by (J2) we have 1 J( 2, 4 ) and 4 J( 1, 5 ). Since 1 4, we have, by (J3), that 1 J( 2, 5 ). Similarly we have 2 J( 1, 5 ), which violates (b1). If G J contains a long cycle, say C = 1 2 n 1 with n 5 as an indced sbgraph, then, by applying (J2) and (J3) sccessively, we get that 2 J( 1, n 1 ) and 1 J( 2, n 1 ), which violates (b1). Similarly, if G J has a P as an indced sbgraph, then we can also derive a contradiction. For, let the vertices of the indced P-graph be as in Fig. 1. By (J2) we have w J(, x), x J(w, y) and y J(x, v). Applying (J3) on x J(y, w), w J(x, ), we get x J(, y). Now applying (J3) on x J(, y), y J(x, v), we get x J(, v). By a similar argment it follows that v J(, x), which contradicts (b1). Lemma 3. Let J be a transit fnction on a non-empty finite set V satisfying the axioms (b1), (J2), (J2 ) and (J3 ) with nderlying graph G J. Then G J is HHD-free. Proof. First recall that J(, v) = {, v}, for any edge v in G J. Sppose G J contains a hose as an indced sbgraph with vertices shown in Fig. 1(A). Then, by (J2), we have 1 J( 2, 4 ), 4 J( 1, 5 ). By definition of G J, we have J( 2, 1 ) = J( 1, 4 ) = J( 4, 5 ) = 2 with 1 4, 2 5. Hence by (J2 ) we have 1 J( 2, 5 ). Similarly, 2 J( 1, 5 ), which violates (b1). If G J contains a long cycle, say C = 1 2 n 1 with (n 5) as an indced sbgraph, then, by applying (J2), (J2 ) and (J3 ) sccessively, we get that 2 J( 1, n 1 ) and 1 J( 2, n 1 ) which violates (b1). Assme that G J contains a domino as an indced sbgraph, (say) with vertices 1, 2, 3, 4, 5, 6 as shown in Fig. 1(B). Here also sing (J2), (J2 ) and (J3 ) we get that 3 J( 1, 4 ) and 4 J( 1, 3 ), which violates (b1). Lemma 4. Let J be a transit fnction on a non-empty finite set V satisfying the axioms (b1), (b2), and (J1) with nderlying graph G J. If w J(, v), w, v, then there exists a seqence 1, 2,..., k V satisfying the conditions (i) J( i+1, ) J( i, ), i = 0, 1, 2,..., where 0 = w, k+1 =, (ii) i J( i 1, i+1 ), i = 1, 2, 3,..., k sch that w, 1, 2,..., k, is a path in G J. (iii) i i+1 E(G J ), i = 0, 1, 2, 3,..., k. Proof. Since w J(, v) and w is not eqal to and v, we can se (J1) to find a neighbor 1 of w in J(, w). Using (b1) and (b2), it follows that J(, 1 ) is a proper sbset of J(, w). If J(, 1 ) > 2, we can se (J1) to find a neighbor 2 of 1 in J(, 2 ). Using again (b1) and (b2), it follows that J(, 2 ) is a proper sbset of J(, 1 ). If J(, 2 ) > 2, we contine the argment. Since this process cannot contine indefinitely, we end p either with 1 =, in which case we take k = 0, or there exists a positive k sch that J(, k 1 ) > 2 bt J(, k ) 2. Note that in the latter case k, being adjacent to k 1, mst be distinct from. So J(, k ) = 2. Now we take k+1 =. Ths we get a seqence of vertices w, 1, 2, 3,... sch that (i) J( i+1, ) J( i, ) sing (b1) and (b2), i = 0, 1, 2,..., k with 0 = w, (ii) i J( i 1, i+1 ), i = 1, 2, 3,..., k (iii) i i+1 E(G J ), i = 0, 1, 2, 3,..., k.

6 M. Changat et al. / Discrete Applied Mathematics 158 (2010) Now in (i), (ii) and (iii) above we have i k. Finally, from (iii) above we dedce that w, 1, 2,..., k, is a w, -walk in G j. Take i, j with i < j, then j J( i, ), by (i) and i J( j, ) by (b1). So i j, whence the walk is a path, and we are done. Now we are ready for the main reslts of this paper: the characterization of transit fnction in terms of transit axioms only that are precisely the indced path fnction of some graph. Becase of Nebeský s impossibility reslt in [23], we have to restrict orselves to special instances. In or case this means that we restrict orselves to transit fnctions that are a betweenness and satisfy some additional axioms. Note that the restrictions on J imply restrictions on the graph as well. These restrictions do not appear explicitly in the statements of the theorems, bt they follow trivially from Lemmas 2 and 3. We present two sch characterizations. Theorem 2. Let V be a finite non-empty set and J be a transit fnction on V satisfying the axioms (b1), (b2), (J1), (J2) and (J3). Let G J be the nderlying graph of the transit fnction J. Then J is precisely the indced path fnction of G J. Proof. Let and v be two distinct vertices of G J, and let w be a vertex in J(, v). We have to prove that w lies on some indced, v-path. Since G J is connected, there is at least one indced, v-path. Hence w J(, v) whenever w = or w = v. So let s assme that w, v. Now we apply Lemma 4. There exists a, w-path P = k k+1 with w = 0 and = k+1 satisfying (i) J( i+1, ) J( i, ), i = 1, 2, 3,..., k, (ii) i J( i 1, i+1 ), i = 1, 2, 3,..., k, (iii) i i+1 E(G J ), i = 1, 2, 3,..., k. From (J1) we also dedce the existence of a neighbor v 1 of w in J(v, w) \ J(, w) sch that w J( 1, v 1 ). Applying Lemma 4 on J(w, v) and v 1 we get in a similar way a v, w-path P v = v 0 v 1 v 2... v k v k +1 with w = v 0 and v = v k +1 satisfying conditions similar to (i), (ii) and (iii). Claim 1. P is an indced, w-path and P v is an indced w, v-path. We need to prove that i i+l E(G J ), for i = 0, 1, 2,..., k l with l 2. When l = 2, the reslt follows by (ii). In the case l = 3, assme the contrary, that is i i+3 E(G J ). Then, by (J2), we have i J( i+1, i+3 ). By (i), we have J( i+3, ) J( i+2, ) J( i+1, ). That is, i+3 J( i+3, ) J( i+1, ). Therefore by (b2), we have J( i+1, i+3 ) J( i+1, ). Ths we get i J( i+1, ). On the other hand, by (i) we have i+1 J( i, ). Bt now we get a conflict with (b1). Hence i i+3 E(G J ). Since indced long cycles are forbidden by Lemma 2, we infer that P is indced. Similarly, P v is indced, and Claim 1 follows. Claim 2. i and v j are not adjacent, for any i and j. Since w J( 1, v 1 ), we have 1 v 1 E(G J ). Sppose that 1 v 2 E(G J ), then by (J2), we wold have 1 J(w, v 2 ), since wv 2 E(G J ) and 1 w E(G J ). Note that, by (b2) we have J(w, v 2 ) J(w, v). So we wold have 1 J(w, v), whereas by applying (J1), we have already 1 J(w, v), which is a contradiction and therefore 1 v 2 E(G J ). Similarly v 1 is not adjacent to 2. Now sppose that there is a vertex i adjacent to a vertex v j, for some i and j. Then we chose i and j as small as possible. The previos argment tells s that then w P i v j v j P v w is an indced cycle of length at least 5. This impossibility settles Claim 2. Next we prove that no vertex in 1, 2,..., k+1 coincides with a vertex in v 1, v 2,..., v k +1. Evidently 1 v 1. Sppose i = v j, for some i and j. Withot loss of generality we may assme that i j. Note that we have i > 1. Then i 1 is adjacent to v j, which is a contradiction by Claim 2. Hence P P v is an indced, v-path and w lies on it. Ths we have shown that J(, v) J GJ (, v). To complete the proof we show that J GJ (, v) J(, v), that is, for any vertex w on some indced, v-path P, we prove that w J(, v). This is done by indction on the length l(p) of P. If w = or v, then evidently w J(, v). Therefore assme that w, v, so that l(p) 2. When l(p) = 2, the reslt follows by (J2). Assme that the reslt is tre for l(p) < m. Sppose now that l(p) = m with m > 2. Then, either or v has a neighbor on P different from w, say. Let be the neighbor of on P. So lies on the indced w, -sbpath of P and w lies on the indced v, -sbpath of P. By the indction hypothesis we have w J(v, ) and J(w, ), hence by (J3) with x = w and y = we have w J(v, ). Since J is a transit fnction it follows that w J(, v). It trns ot that in the above theorem we can replace axiom (J3) by the two axioms (J2 ) and (J3 ). Theorem 3. Let V be a finite non-empty set and J be a transit fnction on V satisfying the axioms (b1), (b2), (J1), (J2), (J2 ), (J3 ). Let G J be the nderlying graph of the transit fnction J. Then J is precisely the indced path fnction of G J. Proof. Let and v be two distinct vertices of G J, and let w be a vertex in J(, v). The proof that w lies on some indced, v-path is similar to the proof of the same assertion in the previos Theorem. The only difference is that where we applied Lemma 2 we now apply Lemma 3.

7 432 M. Changat et al. / Discrete Applied Mathematics 158 (2010) To complete the proof we show that J GJ (, v) J(, v), that is, for any vertex w on some indced, v-path P, we prove that w J(, v). Again this is done by indction on the length l(p) of P. The cases when w = or v, and when l(p) = 2, are also the same as that in the proof of the previos Theorem. So assme that l(p) 3. If l(p) = 3, the reslt follows by (J2 ). Sppose l(p) = 4. If w is adjacent to or v, say, then let v be the neighbor of v on P. Since the reslt holds for indced paths of length 3, we have w J(, v ) and v J(w, v). Note that w is not adjacent to v, hence J(w, v ) {w, v }. Therefore, by (J3 ) with x = w and y = v, we have w J(, v). If w is not adjacent to or v, then, by the previos argment, J(, v). where is the vertex adjacent to on P. Therefore J(, v) J(, v) by (b2). Also w J(, v), since the reslt is tre for l(p) = 3. Hence w J(, v). Assme that the reslt is tre for l(p) < m. Let l(p) = m with m > 4. Consider the case when w is adjacent to either or v. Let s assme that w is adjacent to. Let v be the neighbor of v on P. Since m > 4, J(w, v ) {w, v }. Also by the indction hypothesis, w J(, v ) and v J(w, v). Hence, by (J3 ) with y = v, we have w J(, v). Finally, consider the case when w is not adjacent to or v. Let be the neighbor of on P, and let v be the neighbor of v on P. Then, becase of l(p) 5, vertex w cannot be adjacent to both and v, say w is not adjacent to. By the indction hypothesis, we have w J(, v ) and J(, w). Then, by (J3 ) with x = w and y = v, we get w J(v, ). Since J is a transit fnction this implies w J(, v). The aim of the above theorems was to obtain instances where a set of transit axioms characterizes the transit fnction as the indced path fnction of some graph. By Lemma 2 we know that in Theorem 2 the graphs involved have to be HHP-free, and by Lemma 3 we know that in Theorem 3 the graphs involved have to be HHD-free. Bt the theorems do not yet specify on what graphs the indced path fnction actally satisfies the respective sets of axioms. The following theorem is in a way the converse of Theorem 2. Theorem 4. Let G = (V, E) be a connected HHP-free graph, and let J be the indced path fnction of G. Then J satisfies the axioms (b1), (b2), (J1), (J2), and (J3). Proof. An indced path fnction always satisfies (J2). Since G is HHP-free, G is HHD-free. By Theorem 1 and Corollary 1, J satisfies (b1), (b2), and (J1). Sppose that J does not satisfy (J3). Then we can find distinct vertices, x, y, v with and v non-adjacent sch that x J(, y), y J(x, v) whereas x J(, v). Let P be an indced, y-path throgh x, and let Q be an indced x, v-path throgh y. Going from x to v along Q, let y be the last vertex that is still on (x) P y. Note that, if x and y are adjacent, then y = y. Otherwise y is a vertex in (x) P y not adjacent to x. Let z be the next vertex on Q after y, so that z is not on x P y. Then z cannot be on P (x), for otherwise the edge y z wold be a chord of P. So z is not on P. Note that, Q being indced, z and x are not adjacent. Let p be the vertex on (x) P y closest to x that has a neighbor on y Q v, and let q be sch a neighbor closest to v. Then q is not on P (x), for otherwise pq wold be a chord of P. Note that q might be z, and p might be y, in which case q mst be z. We write Q = p q Q v. Consider the, v-walk R = P x P p Q v. First we show that R is a path. If not, then the sbwalks P x and p Q v mst have a vertex in common. Let s sch a vertex on (q) Q v closest to q. Then s is on P (x). Now C = s P x P p q Q s is a cycle containing the for distinct vertices x, p, q, s. Note that s and x cannot be adjacent, for otherwise sx wold be a chord of Q. Let t be the neighbor of x on s P x and let t be the neighbor of t on C distinct from x. So C contains at least five distinct vertices. To avoid a hole C cannot be indced. Let q s be the chord from q Q s to s P x with q closest to q and then s closest to t. Then s P x P p q... Q q s is an indced cycle of length at least for, hence a 4-cycle. This is only possible if s = t, and qt and xp are edges. Now t is a vertex adjacent to t and possibly also to q bt not to x or p. So we have either an indced P-graph or an indced hose. Ths we conclde that the, v-walk R actally is a, v-path. Since x J(, v), the path R cannot be indced, and there mst be chords. Becase of the choice of p and q, any sch chord mst be between q Q v and P (x). Take any sch chord ts with t on Q closest to q and then s on P closest to x. Now x, p, t, s are for distinct vertices, and the cycle C = x P p Q t s P x is an indced cycle. To avoid a hole it mst be of length for. Recall that and v are not adjacent. So we cannot have both = s and v = t. Therefore we can find a vertex on P (s) adjacent to s or a vertex on (t) Q v adjacent to t. It is easy to see that any sch vertex can only be adjacent to t and/or s bt not to x or p. So, together with C this vertex forms either an indced hose or an indced P-graph. This is impossible, and the proof is complete. The converse of Theorem 3 is the following reslt. Theorem 5. Let G = (V, E) be a connected HHD-free graph, and let J be the indced path fnction of G. Then J satisfies the axioms (b1), (b2), (J1), (J2), (J2 ), and (J3 ). Proof. Note that the only difference between (J3) and (J3 ) is that in (J3 ) the vertices x and y cannot be adjacent. Otherwise the proof is identical to that of Theorem 4.

8 M. Changat et al. / Discrete Applied Mathematics 158 (2010) The monotonicity axiom (m) is stronger than the betweenness axiom (b2). So the class of graphs on which the indced path fnction is monotone mst be more restricted. Corollary 2 provides s with proofs of the next two theorems. We omit the details. Theorem 6. Let G = (V, E) be a connected graph that is HHP-free, K 2,3 -free and W 4 e-free, and let J be the indced path fnction G. Then J satisfies (b1), (m), (J1), (J2), and (J3). Theorem 7. Let G = (V, E) be a connected graph that is HHD-free, K 2,3 -free, W 4 e-free, and let J be the indced path fnction G. Then J satisfies (b1), (m), (J1), (J2), (J2 ), and (J3 ). 4. Conclsion The geodesic interval fnction I allows an axiomatic characterization in terms of transit axioms, as was first shown by Nebeský in [20]. On the other hand Nebeský showed in [23] that the indced path fnction J does not allow sch an axiomatic characterization. This posed the challenge to search for special instances where it wold still be possible to have sch axiomatic characterizations. In [16] the indced path fnction was stdied from the perspective of betweenness: the indced path fnction J of a connected graph G satisfies the two simple betweenness axioms (b1) and (b2) if and only if G is HHD-free. In this paper we extend this reslt and obtain two instances of an axiomatic characterization of the indced path fnction. The main reslts are of the type: if a transit fnction J is a betweenness and satisfies some extra axioms, then it is the indced path fnction of some graph. In some sense these reslts provide s with a bons, viz. a stronger reslt than the one on betweenness in [16]: if G is HHD-free then its indced path fnction is not only a betweenness bt satisfies also the other extra axioms. We also presented some related reslts involving monotonicity, another well stdied transit axiom. 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