Transit functions on discrete structures were introduced by Mulder muld-08 to generalize some basic notions in discrete geometry, amongst which convexity, interval and betweenness. A transit function on a non-empty set is a function to on satisfying the following three axioms:
, for all ,
, for all ,
, for all .
If is the vertex set of a graph , then we say that is a transit function on . Throughout this paper, we consider only finite, simple and connected graphs. The underlying graph of a transit function on is the graph with vertex set , where two distinct vertices and are joined by an edge if and only if .
A - shortest path in a connected graph is a -path in containing the minimum number of edges. The length of a shortest -path (that is, the number of edges in ) is the standard distance in . The interval function of a connected graph is the function defined with respect to the standard distance in as
: lies on some shortest - path in
The interval function is a classical example of a transit function on a graph ( we some times denote by , if there is no confusion for the graph ). It is easy to observe that the underlying graph of is isomorphic to . The term interval function was coined by Mulder in mu-80 , where it is extensively studied using an axiomatic approach.
Nebeský initiated a very interesting problem on the interval function of a connected graph during the 1990s. The problem is the following: “ Is it possible to give a characterization of the interval function of a connected graph using a set of simple axioms (first - order axioms) defined on a transit function on ?” Nebeský nebe-94 ; nebesky-94 proved that there exists such a characterization for the interval function by using first - order axioms on an arbitrary transit function . In further papers that followed nebe-95 ; ne-08 ; nebesky-08 ; nebe-01 , Nebeský improved the formulation and the proof of this characterization. Also, refer Mulder and Nebeský mune-09 . In chfermuna-18 , the axiomatic characterization of is extended to that of a disconnected graph. In all these characterizations, five essential axioms known as classical axioms are always required. These five classical axioms are and and three additional , , and defined as follows:
|if and , then ,|
|if and then ,|
The notation can be interpreted as is in between and . For example, the axiom can be interpreted as: if is between and , and is between and , then is between and . Similarly we can describe all other axioms. Hence we use the terminology of betweeness for an axiom on a transit function . The above interpretation was the motivation for the concept of betweenness in graphs using transit functions. It was formally introduced by Mulder in
muld-08 as those transit function that satisfy axioms and . Here the axiom is defined for every and a transit function
The following implications can be easily verified for a transit function among axioms and .
Axioms and implies axiom .
Axioms and implies axiom which implies axiom ( that is , for a transit function implies implies )
The converse of the above implications need not hold. A transit function satisfying axioms and is known as a geometric transit function.
The problem of characterizing the interval function of an arbitrary graph can be adopted for different graph classes; viz., characterizing the interval function of special graph classes using a set of first - order axioms on an arbitrary transit function. Such a problem was first attempted by Sholander in Sholander-52 with a partial proof for characterizing the interval function of trees. Chvátal et al. Chvatal-11 obtained the completion of this proof. Further new characterizations of the interval function of trees and block graphs are discussed in kaMc-01 . Axiomatic characterization of the interval function of median graphs, modular graphs, geodetic graphs, (claw, paw)-free graphs and bipartite graphs are respectively described in chferna-16 ; mfn ; mu-80 ; mune-09 ; nebe-95 .
In this paper, we continue the approach of characterizing the interval function of some related classes of graphs, namely, distance hereditary graphs, Ptolemaic graphs and bridged graphs. We fix the graph theoretical notations and terminology used in this paper. Let be a graph and a subgraph of . is called an isometric subgraph of if the distance between any pair of vertices, in coincides with that of the distance . is called an induced subgraph if are vertices in such that is an edge in , then must be an edge in also. A graph is said to be -free, if has no induced subgraph isomorphic to . Let be graphs. For a graph , we say that is -free if has no induced subgraph isomorphic to , . Chordal graph is an example of a graph which is defined with respect to an infinite number of forbidden induced subgraphs ( is chordal if have no induced cycles with length more than three). There are several graphs that can be defined or characterized by a list of forbidden induced subgraphs or isometric subgraphs. See the survey by Brandstädt et al. brandstadt1999graph and the information system isgci , for such graph families. A graph is a bridged graph if has no isometric cycles of length greater than . Clearly the family of bridged graphs contain the family of chordal graphs. The graph is distance hereditary if the distances in any connected induced subgraph of are the same as in . Thus, any induced subgraph inherits the distances of . The graph is a Ptolemaic graph if is both chordal and distance hereditary. Both Ptolemaic graphs and distance hereditary graphs possess a characterization in terms of a list of forbidden induced subgraphs, while bridged graphs by definition itself possess an infinite list of forbidden isometric subgraphs. In this paper, our idea is to find suitable axioms that fail on every forbidden induced subgraph for the Ptolemaic and distance hereditary graphs, while that for the bridged graph is to find an axiom that fails on all of its forbidden isometric subgraphs, namely on all isometric cycles .
In addition to the geometric axioms and , we consider the following betweenness axioms and for a transit function on for proving the characterizations of these classes of graphs.
: For any pair of distinct vertices we have .
From the definition of the axioms, we observe the following. The axiom is a simple betweenness axiom which is always satisfied by the interval function . The axiom is a natural extension of . We provide examples in the respective sections for the independence of the axioms and and . The axioms and were first considered in mcjmhm-10 and later in Changat-22 . The axiom first appeared in Sholander-52 for characterizing the interval function of trees. The axiom is discussed in Changat-22 for characterizing the interval function of a Ptolemaic graph . We may observe that both the family of bridged graphs and distance hereditary graphs is a strict super class of the family of Ptolemaic graphs,, that is, and and and coincide only in . But, and , This relation is also reflected in the implications between the axioms and . From the definitions, we have the axiom implies axioms and , and also implies , while the reverse implications are not true. In other words, axioms and are weaker axioms than and hence graphs whose interval function satisfy axioms and will be a super class of graphs whose interval function satisfies . Similarly graphs whose interval function satisfies axiom will be a super class of graphs whose interval function satisfies . See Figure 1 for the relationships between the family , and .
We organize the results as follows. In Section 2, we characterize the interval function of a distance hereditary graph, Ptolemaic graph in Section 3, bridged graph in Section 4 and in Section 5, a discussion of the so called induced path transit function for the distance hereditary graphs respectively.
2 Interval function of Distance hereditary graphs
For the axiomatic characterization of the interval function of distance hereditary graphs, we require the axioms and . First we show that these axioms are independent with the Examples 1 and 2 below. Also it is clear from the Figures 2 and 3 that the axioms and are independent. The interval function doesn’t satisfy axiom , while satisfy axiom for the graphs in Figure 2. Also doesn’t satisfy axiom , while satisfy axiom for the graphs in Figure 3. By an - fan - free graph , we mean that is free from the House graph, the Hole graph (cycles , ), the Domino graph and the -fan (See the Figures 2 and 3 for these graphs).
Example 1 ( but not ).
Let . Define a transit function on as follows. and . We can easily see that satisfies . But we can see that and but . So does not satisfy .
Example 2 ( but not ).
Let . Define a transit function on as follows: and . We can see that satisfies . But , but . Hence does not satisfy .
Changat-22 For every graph , satisfies the axiom if and only if is house, , -fan free.
mcjmhm-10 Let be a transit function on a non-empty finite set satisfying the axioms and with underlying graph . Then is -free.
Bandelt and Mulder obtained a forbidden induced subgraph characterization of distance hereditary graphs in bamu-86 . We quote the theorem as
bamu-86 A graph is distance hereditary if and only if is - fan-free.
We state a related result from Changat-22 using the axiom defined for a transit function as
Note that a graph is the graph obtained by adding a pendent edge on an induced 4-cycle, . It follows from the definition that axiom implies both the axioms and , but the reverse implications are not true. We quote the result.
Changat-22 For every graph , satisfies the axiom if and only if is - fan - free graph.
It may be observed that a graph is a distance hereditary graph and hence the class of - fan - free graphs is a proper subclass of the class of - fan - free graphs (distance hereditary graphs). The proof of the next theorem characterizing the class of distance hereditary graphs follows the same lines of ideas as in the proof of Theorem 2 with modifications since the axioms and are weaker axioms than the axiom .
Let be a connected graph. The interval function satisfy the axioms and if and only if is a distance hereditary graph.
We use the fact from Theorem 1 that distance hereditary graphs are precisely - fan - free graphs for the proof.
Suppose that the interval function of satisfy the axioms and . To prove that is - fan - free, assume the contrary that contains a house, a hole, or a domino or a -fan as an induced subgraph. A graph with an induced house or - fan or a
doesn’t satisfy ( The vertices in Figure 2 doesn’t satisfy the axiom ).
For an isometric hole , , we choose vertices as shown in Figure
3, to prove that is violated. If has a domino ,
which is an isometric subgraph of with vertices as shown in Figure 3, then . If is not isometric, then there is
a vertex adjacent to both and or and . In this case, the graph
induced by is either a or a house or a -fan.
Let be a hole in that is not isometric and assume that is minimum. Clearly and there exist such that . Let be a -geodesic; we may choose such that is minimum. Let and be the -paths on where , and and . Clearly and and we may assume without loss of generality that . Moreover, we can choose such that the cycle induced by has minimum length. In particular, by the choice of , is not adjacent to any of whenever . By minimality of , , can be adjacent to (or ) only if or . If , then form (together with possibly some additional vertices of or ) an induced hole shorter than , which is not possible. If , then and must be adjacent; otherwise we have a shorter hole than on vertices together with or (and possibly some other vertices of or ). But then form a -cycle and we have an induced domino on when and an induced house on the same vertices when . Let now . First note that must be adjacent to at least one of and of by the minimality of , since there are no isometric holes. Let first . If either or is not adjacent to , we have an induced house as a subgraph. Otherwise we have an induced - fan on . Let now . By the above, is not adjacent to or to and is not adjacent to or . Suppose that at least one of or , say , is adjacent to . If both and are adjacent to , we have an induced - fan on . If is adjacent to but is not, we have an induced house on . If both and are not adjacent to , adjacent to and adjacent to , we have an induced domino on . Hence but and by a similar argument but . Since , and must be adjacent since there are no isometric holes or by the minimality of . This yields an induced house on . Finally, If the induced is not isometric with , the only case left is the one in Figure 3, and by choosing the vertices as in the figure, it follows that the axiom is violated. Therefore, when has an induced house, hole, domino or - fan, either the axiom or is violated.
Conversely assume that axiom or is not satisfied by the interval function of . It is already known from Proposition 1 that if axiom is not satisfied, then has an induced , House or - fan. Now suppose axiom is not satisfied. Then there exists distinct vertices in such that , , and . Let be a -geodesic containing and be a - geodesic containing . We claim that
is a -path.
Since we see that is a geodesic. Therefore , for otherwise we may find a shorter path from to .
Now we claim that . Assume on the contrary, that , and let be the last vertex of the intersection (when going from to along path). Then , and since we find that . Hence we have that the path is shorter than , a contradiction.
Hence is a -path. Since , is not an geodesic. If is any
-geodesic, then . Fix an -geodesic . Let
be the last vertex on before that is on and be
the first vertex on after that is on . Note that such
vertices always exists, since and . On
the other hand, note that can be equal to , but .
Label vertices of the path by . Label vertices of
the -subpath of as . Clearly
and and . Path is not necessarily an induced path. If not, we choose
among all chords the one with maximal and replace the
part by this chord. Vertices of this new path are
denoted by , where still by
the choice of and . But is an induced
path, since it is a shortest path. Note that is not adjacent
to by the choice of . Hence can be
adjacent only to . Similarly , for , cannot be
adjacent to . We consider the following two
Case 1: .
If also , we have an induced cycle of length . So let . Also and , otherwise we have a house on vertices , and or respectively. This implies that is an edge and we have an induced domino (), otherwise we have an induced cycle of length .
Case 2: .
If , we have a house if or a - fan on vertices , and . Hence and also . We get an induced cycle of length , if is not adjacent to at least one of or . If first and , we get a house on vertices , and . Let now and . Since , there exists . If , we get a house on vertices , and . Also , since otherwise we get a house on vertices , and . But now we have an induced path which lead to an induced hole. Finally, if and , we get a - fan on vertices , and . Thus in all cases, we get either an induced house, hole, domino, or - fan, and thus the proof is completed. ∎
We need the following Lemma
Let be a transit function on a non-empty finite set satisfying the axioms and with underlying graph . Then is - fan - free.
Since for a transit function axiom implies axiom , by Lemma 1, is -free. We prove that is also - fan - free. Suppose on the contrary, contains a - fan with vertices as an induced subgraph. Let be the path of length three and be the vertex adjacent to all of . Since and , we have by axiom , and . Again since by . Again, and , , by axiom , we have . Now, we have and . Hence by axiom , we have , which is a contradiction and hence the lemma is proved. ∎
3 Axiomatic characterization of the interval function of Ptolemaic graphs
For the axiomatic characterization of of a Ptolemaic graph , the essential axiom is . Ptolemaic graphs are chordal graphs that are - fan - free. Changat et al. in Changat-22 characterized the graphs for which the interval function satisfies the axiom as follows.
Changat-22 Let be a graph. The interval function satisfies the axiom if and only if is a Ptolemaic graph.
Let be any transit function defined on a non-empty set . If satisfies and then the underlying graph of is -free for .
Let be a transit function satisfying and . Let contains an induced cycle say . Without loss of generality assume is the minimum such cycle (in the sense that length of the induced cycle is as small as possible). We prove for every .
Now since and , By axiom we have in a similar fashion we can show that . Since satisfies (J0)-axiom we have , which is a contradiction as .
As in the above case, we can see that , this together with we have , which is again a contradiction.
By repeated application of -axiom, as in the above two cases, with , by applying , we can see that . Here again a contradiction.
Hence, in all cases, we can see that does not contain as an induced subgraph This completes theorem. ∎
The following straightforward Lemma for the connectedness of the underlying graph of a transit function is proved in mcjmhm-10 .
mcjmhm-10 If the transit function on a non-empty set satisfies axioms and , then the underlying graph of is connected.
We have the following lemma.
If is a transit function on satisfying the axioms and , then satisfies axiom and is connected.
Let satisfies axioms and . To prove satisfies . Since satisfies , For , let , and . Since satisfies , we have . Now and so by axiom , we have , which implies that satisfies . Connectedness of follows from Lemma 3, since satisfies axioms and as axiom implies axiom . ∎
Example 3 ( and but not ).
Let . Let be defined as follows. and in all other cases . Since is a - fan, satisfies and . Next to show satisfies axiom. Since , we can see that for all so that for this pair satisfies axiom. Now consider , we can see that we have and which implies that satisfies axiom for this pair too. The case is similar for . All other pairs corresponds to edges. Hence we can see that satisfies -axiom. Now but , and violates axiom.
Let be any transit function satisfying the axioms and then is Ptolemaic and .
Since satisfies the axioms and , we have that is a chordal graph by Theorem 5. To prove that is Ptolemaic, we have to show that is - fan - free. Suppose that contains an induced - fan with vertices with forming a path and as the vertex adjacent to all the vertices . Since and are adjacent and is not an edge, by , . Similarly . Since is a transit function, by , and and hence by , . Again, since and are edges and is not an edge, . That is, and , by , we have , which is not true as is an edge. That is, we have proved that is a chordal graph which is - fan - free and hence is a Ptolemaic graph. By Lemma 4, satisfies axiom and and is connected.
Now we prove that for all . We prove by induction on the distance between and .
Case when .
Let Hence we can see that . That is, and , since satisfies . Therefore . Conversely suppose . Suppose . Since there exists at least one element such that are edges in . By assumption, is not adjacent to both and . Assume that is not an edge. Since and satisfies and with . By applying axioms and continuously on , we get vertices such that and , for and since is finite, , for some , say . That is, we have vertices