m_{3}^{3}-Convex Geometries are A-free.

# m33-Convex Geometries are A-free.

## Abstract

Let be a finite set and a collection of subsets of . Then is an alignment of if and only if is closed under taking intersections and contains both and the empty set. If is an alignment of , then the elements of are called convex sets and the pair is called an aligned space. If , then the convex hull of is the smallest convex set that contains . Suppose . Then is an extreme point for if . The collection of all extreme points of is denoted by . A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let be a connected graph and a set of vertices of . A subgraph of containing is a minimal -tree if is a tree and if every vertex of is a cut-vertex of the subgraph induced by . The monophonic interval of is the collection of all vertices of that belong to some minimal -tree. A set of vertices in a graph is -convex if it contains the monophonic interval of every -set of vertices is . A set of vertices of a graph is -convex if for every pair of vertices in , the vertices on every induced path of length at least 3 are contained in . A set is -convex if it is both - and - convex. We show that if the -convex sets form a convex geometry, then is -free.

Key Words: minimal trees, monophonic intervals of sets, -monophonic convexity, convex geometries
AMS subject classification: 05C75, 05C12, 05C17

## 1 Introduction

Let and be graphs. Then is an induced subgraph of if is a subgraph of and for every , if and only if . We say a graph is -free if it does not contain as an induced subgraph. Suppose is a collection of graphs. Then is -free if is -free for every . If is a path or cycle that is a subgraph of , then has a chord if it is not an induced subgraph of , i.e., has two vertices that are adjacent in but not in . An induced cycle of length at least is called a hole.

Let be a finite set and a collection of subsets of . Then is an alignment of if and only if is closed under taking intersections and contains both and the empty set. If is an alignment of , then the elements of are called convex sets and the pair is called an aligned space. If , then the convex hull of is the smallest convex set that contains . Suppose . Then is an extreme point for if . The collection of all extreme points of is denoted by . A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. This property is referred to as the Minkowski-Krein-Milman () property. For a more extensive overview of other abstract convex structures see [13]. Convexities associated with the vertex set of a graph are discussed for example in [3]. Their study is of interest in Computational Geometry and has applications in Game Theory [2].

Convexities on the vertex set of a graph are usually defined in terms of some type of ‘intervals’. Suppose is a connected graph and two vertices of . Then a geodesic is a shortest path in . Such geodesics are necessarily induced paths. However, not all induced paths are geodesics. The -interval (respectively, -interval) between a pair of vertices in a graph is the collection of all vertices that lie on some geodesic (respectively, induced path) in and is denoted by (respectively, ).

A subset of vertices of a graph is said to be -convex (-convex) if it contains the -interval (-interval) between every pair of vertices in . It is not difficult to see that the collection of all -convex (-convex) sets is an alignment of . A vertex is an extreme point for a -convex (or -convex) set if and only if is simplicial in the subgraph induced by , i.e., every two neighbours of in are adjacent. Of course the convex hull of the extreme points of a convex set is contained in , but equality holds only in special cases. In [6] those graphs for which the -convex sets form a convex geometry are characterized as the chordal -fan-free graphs(see Fig. 1). These are precisely the chordal, distance-hereditary graphs (see [1, 7]). In the same paper it is shown that the chordal graphs are precisely those graphs for which the -convex sets form a convex geometry.

For what follows we use to denote an induced path of order . A vertex is simplicial in a set of vertices if and only if it is not the centre vertex of an induced in . Jamison and Olariu [8] relaxed this condition. They defined a vertex to be semisimplicial in if and only if it is not a centre vertex of an induced in .

Dragan, Nicolai and Brandstädt [5] introduced another convexity notion that relies on induced paths. The -interval between a pair of vertices in a graph , denoted by , is the collection of all vertices of that belong to an induced path of length at least . Let be a graph with vertex set . A set is -convex if and only if for every pair of vertices of the vertices of the -interval between and belong to . As in the other cases the collection of all -convex sets is an alignment. Note that an -convex set is not necessarily connected. It is shown in [5] that the extreme points of an -convex set are precisely the semisimplicial vertices of . Moreover, those graphs for which the -convex sets form a convex geometry are characterized in [5] as the (house, hole, domino, )-free graphs (see Fig. 1).

More recently a graph convexity that generalizes -convexity was introduced (see [11]). The Steiner interval of a set of vertices in a connected graph , denoted by , is the union of all vertices of that lie on some Steiner tree for , i.e., a connected subgraph that contains and has the minimum number of edges among all such subgraphs. Steiner intervals have been studied for example in [9, 12]. A set of vertices in a graph is -Steiner convex (-convex) if the Steiner interval of every collection of vertices of is contained in . Thus is -convex if and only if it is -convex. The collection of -convex sets forms an aligned space. We call an extreme point of a -convex set a -Steiner simplicial vertex, abbreviated vertex.

The extreme points of -convex sets , i.e., the vertices are characterized in [4] as those vertices that are not a centre vertex of an induced claw, paw or , in see Fig. 1. Thus a vertex is semisimplicial. Apart from the -convexity, for a fixed , other graph convexities that (i) depend on more than one value of and (ii) combine the convexity and the geodesic counterpart of the -convexity were introduced and studied in [10]. In particular characterizations of convex geometries for several of these graph convexities are given.

The notion of an induced path between a pair of vertices can be extended to three or more vertices. This gives rise to graph convexities that extend the -convexity. Let be a set of at least two vertices in a connected graph . A subgraph containing is a minimal -tree if is a tree and if every vertex is a cut-vertex of . Thus if , then a minimal -tree is just an induced path. Moreover, every Steiner tree for a set of vertices is a minimal -tree. The collection of all vertices that belong to some minimal -tree is called the monophonic interval of and is denoted by . A set of vertices is -monophonic convex, abbreviated as -convex, if it contains the monophonic interval of every subset of vertices of . Thus a set of vertices in is a monophonic convex set if and only if it is a -convex set. By combining the - convexity with the -convexity introduced in [5], we obtain a graph convexity that extends the graph convexity studied in [10]. More specifically we define a set of vertices in a connected graph to be -convex if is both - and -convex. In this paper we show that if the -convex alignment forms a convex geometry then is -free. We use the fact that these graphs are -free for several other graphs . In particular is easily seen to be house, hole, and domino free. Moreover the graphs of Fig. 2 are forbidden. A graph is a replicated twin if it is isomorphic to any one of the four graphs shown in Fig. 2(a), where any subset of the dashed edges may belong to . The collection of the four replicated twin graphs is denoted by . A graph is a tailed twin if it is isomorphic to one of the two graphs shown in Fig. 2(b) where again any subset of the dotted edges may be chosen to belong to . We denote the collection of tailed twin ’s by .

## 2 m33-Convex Geometries are A-Free

Recall that the graphs for which the -convex sets form a convex geometry are characterized in [5] as the (house, hole, domino, )-free graphs. The proof of this characterization depends on the following useful result also proven in [5]:

###### Theorem 1.

If is a (house, hole, domino, )-free graph, then every vertex of is either semisimplicial or lies on an induced path of length at least between two semisimplicial vertices.

In [5] several ‘local’ convexities related to the -convexity were studied. For a set of vertices in a graph , is where is the collection of all vertices adjacent with some vertex of . A set of vertices in a graph is connected if is connected. The following useful result was established in [5].

###### Theorem 2.

A graph G is (house, hole, domino)-free if and only if is -convex for all connected sets of vertices of .

###### Theorem 3.

If is a graph such that is a convex geometry, then is -free.

###### Proof.

Observe first that is (house, hole, domino, , )-free. Suppose is a house, hole, domino, replicated twin or a tailed twin . Then has at most one vertex. Suppose is a graph that contains as an induced subgraph. Then the set of extreme points of the convex hull of is contained in the collection of vertices of . So the convex hull of the extreme points of the -convex hull of is empty or consists of a single vertex. So in this case the -convex alignment of does not form a convex geometry.

If is a set of vertices of a graph , then .

To show that contains no as an induced subgraph we prove a series of lemmas.

###### Lemma 1.

Suppose is a graph for which is a convex geometry. Then for every , .

###### Proof.

By the above observation is (house, hole, domino, , )-free. If then . So we may assume . If , there is a vertex that lies on an induced path between two vertices of . Among all such induced paths of length at least containing , let be one with a minimum number of edges. Suppose is a path. Clearly ; otherwise, . Let . (Suppose .) Then is not adjacent with two non-adjacent vertices of any induced path; otherwise, lies on an induced path.

Case 1 Suppose and lie on a common induced path . We may assume precedes on such a path. Moreover, we may assume that all internal vertices of are not on . For if , , then either or contains , say the former. Since is an induced path, so is . Hence . Thus must have length at least ; otherwise is adjacent with a pair of nonadjacent vertices of , implying that contains an induced path passing through , contrary to assumption. But then we have a contradiction to our choice of .

Case 2 Suppose and lie on two internally disjoint paths and , respectively. We may assume ; otherwise, we are in Case 1.

We show first that no internal vertex of belongs to or . Suppose some internal vertex of or , say belongs to or . However, no internal vertex of belongs to ; otherwise, either the situation arises that was considered in Case 1 or there is an induced path containing . So we may assume that an internal vertex of lies on . Let be the last such vertex. Then contains and is an induced path between two vertices of that is shorter than . So has length ; otherwise we have a contradiction to our choice of . So must be the path . Since has length at least and by our choice of one of the neighbours of on must be . So one of the configurations shown in Fig. 3 must occur where solid lines are edges and dashed lines represent subpaths of and . We may assume that the configuration in (a) occurs. The argument for the configuration in (b) is similar.

Let . Let and be the neighbours of on and , respectively and and the neighbours of on and , respectively. Let and . Since is connected for , it follows from Theorem 2 that is -convex. Since for , every vertex of is adjacent with a vertex of for . In particular is adjacent with a vertex of for . However, is not adjacent with a pair of nonadjacent vertices of nor a pair of nonadjacent vertices of . So without loss of generality we may assume that is adjacent with a vertex of and a vertex of . Also is not adjacent with either or ; otherwise, lies on an induced path.

If is adjacent with two non-adjacent vertices of (or if is adjacent with two nonadjacent vertices of ), then ( and , respectively) and (or , respectively) is an induced path between two vertices of that is shorter than and contains . By our choice of this can only happen if has length .

We consider two subcases that depend on the length of .
Subcase 2.1 Suppose has length .
Then or is , say . The case where can be argued similarly. From the above, we may assume that is adjacent with an internal vertex of and an internal vertex of . The only vertex of that can be adjacent with is ; otherwise, lies on an induced path. So . Now it follows that is not adjacent with a vertex of . Thus is a house unless . If , then ; otherwise, or is a house. So is a tailed twin which is forbidden. So this subcase cannot occur.

Subcase 2.2 Suppose has length at least .
By an earlier observation, is not adjacent with a pair of non-adjacent vertices of and is not adjacent with a pair of non-adjacent vertices of . By assumption, is adjacent with an internal vertex of and an internal vertex of . Suppose . So is not adjacent with a vertex of nor a vertex of .

Fact 1 No vertex of is adjacent with a vertex of and no vertex of is adjacent with a vertex of .
Proof of Fact 1.
Suppose some vertex of is adjacent with a vertex of . Let be the largest integer less than such that is adjacent with a vertex of . Let be a neighbour of on closest to on this path. Then is a cycle of length at least . If , then has length at least and three consecutive vertices of are not incident with a chord of the cycle. This implies that has a hole; which is forbidden. So . Clearly . Let . Then is a cycle of length at least . Thus contains an induced cycle of length at least that contains the edge . Since contains no holes, has length or . Since neither nor is adjacent with a vertex of nor a vertex of and as , it is not difficult to see that the vertices of and induce a house or a domino. So no vertex of is adjacent with a vertex of . By an identical argument we can show that no vertex of is adjacent with a vertex of .

Fact 2 No vertex of is adjacent with any vertex of .
Proof of Fact 2.
Let be the first vertex of that is adjacent with some vertex of . Let be a neighbour of on that is closest to . Then the path is an induced path. So is -convex and hence contains all induced paths of length at least . Since , and since both and contain an internal vertex adjacent with , both and have length at least . So contains all the vertices of and and hence and . So also contains . Thus every vertex of is adjacent with a vertex of or with a vertex of . But by assumption is adjacent with an internal vertex of both and . So is adjacent with a pair of non-adjacent vertices of or a pair of non-adjacent vertices of , neither of which is possible.

From Facts and , it follows that no vertex of the path is adjacent with a vertex of the path . Hence the subgraph induced by the path is an induced path that contains ; contrary to the assumption that . This completes the proof of Case 2.

Case 3 Suppose that belongs to an induced path and to an induced path where and intersect at vertices other than and . We may assume that and do not both belong to nor both to ; otherwise, Case 1 occurs. Let be the last vertex prior to on that is also a vertex of (perhaps ). Let be the first vertex after on that belongs to . So . Let be the last vertex prior to on that also belongs to and the first vertex after on that also belongs to . So .

Subcase 3.1 Suppose contains both and . (Note may precede on .) In this case we can apply the argument used in Case 2 with and replaced by and and and replaced by and . Hence this subcase cannot occur.

Subcase 3.2 Suppose does not contain both and . Then and either lie on or on . We will assume the former case occurs. The arguments for the latter case are similar. We may assume precedes on . The case where precedes on is similar. First suppose that has length . Then is the only interior vertex of and is adjacent with two nonadjacent vertices of . Let be the first vertex on that is adjacent with , and the last vertex of adjacent with . Since , . If precedes on , then the path obtained by taking followed by and then is an induced path that contains both and . Thus we can apply the argument used in Case 1 to this path to obtain a contradiction. If follows on , then we can use the path and the path and apply the argument used in Case 2 with and instead of and , respectively.

We now assume that has length at least . Since is connected it follows, from Theorem 2, that is -convex. Since contains both and it must contain every internal vertex of . So each internal vertex of is adjacent with an internal vertex of . If no internal vertex of is adjacent with a vertex of or , then we can replace in with to obtain an induced path that contains both and . By applying the argument used in Case 1 to this path we obtain a contradiction. Let and be the neighbours of that precede and succeed on . Let be the neighbour of on .