Weakly toll convexity and proper interval graphs

A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.


Introduction
This paper is motivated by the results and ideas contained in (Alcón, Brešar, Gologranc, Gutierrez, Šumenjak, Peterin, and Tepeh, 2015).We introduce a new graph convexity and show how this gives rise to a new structural characterization of proper interval graphs.We begin with an overview of convexity notions in graphs.For an extensive overview of other convex structures, see (van de Vel, 1993).
Let C be a collection of subsets (called convex sets) of a finite set X.In abstract convexity theory, the following axioms determine the pair (X, C) as a convexity space: (i) ∅ and X are convex; (ii) the intersection of any two convex sets is convex.Suppose that X = V (G) for some graph G.For a set S ⊆ V (G), the smallest convex set containing S is called the convex hull of S. A set S ⊆ V (G) is a 2 M.C.Dourado, M. Gutierrez, F. Protti, S. Tondato hull set if the convex hull of S is V (G).An element x ∈ S, where S ⊆ V (G) is a convex set, is called an extreme vertex of S if S \ {x} is also convex.A convex geometry is a pair formed by a graph G and a convexity on V (G) satisfying the Minkowski-Krein-Milman property (Krein and Milman, 1940): Every convex set is the convex hull of its extreme vertices.
The main result of this paper states that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph.In order to prove this result we introduce the concept of weakly toll walk, a walk with a special restriction on their end vertices.A walk u 0 , u 1 , . . ., u k−1 , u k is a weakly toll walk if u 0 u i ∈ E(G) implies u i = u 1 and u j u k ∈ E(G) implies u j = u k−1 .Note that u 1 (or u k−1 ) may appear more than once in the walk.A set S of vertices of G is weakly toll convex if for any two nonadjacent vertices x, y ∈ S, any vertex in a weakly toll walk between x and y is also in S. The weakly toll convexity is the graph convexity defined over weakly toll convex sets.
The concept of weakly toll walk is a relaxation of the concept of tolled walk, which was conceived to capture the structure of the convex geometry associated with an interval graph.Likewise, weakly toll walks are used as a tool to characterize proper interval graphs as convex geometries.
The paper is organized as follows: in Section 2, we give some definitions and necessary background.In Section 3, we prove that proper interval graphs are precisely the convex geometries with respect to the weakly toll convexity.In Section 4, we study some invariants associated with the weakly toll convexity, namely the weakly toll number and the weakly toll hull number.In particular, we determine these invariants in trees and we find bounds in arbitrary interval graphs.Section 5 contains a short conclusion.

Preliminaries
Let G be an undirected graph without loops or multiple edges.If C is a subset of vertices of G, G[C] denotes the subgraph of G induced by C. Let xy ∈ E(G) and z, w be two nonadjacent vertices of G; the graph G−xy+zw is obtained from G by deleting the edge xy and adding the edge zw.
The distance between a pair of vertices u and v of G is the length of a shortest path (or geodesic) between u and v in G, and is denoted by d G (u, v).The geodesic interval I G (u, v) between vertices u and v is the set of all vertices that lie in some shortest path between u and v in G; in other words, x lies in an induced path between u and v} to be the monophonic interval between u and v in G.In the associated monophonic convexity, a subset A tolled walk is any walk T : u 0 , u 1 , . . ., u k−1 , u k such that u 0 is adjacent only to the second vertex of the walk, and u k is adjacent only to the second-to-last vertex of the walk.This implies that each of u 1 and u k−1 occurs exactly once in the walk.
x lies in a tolled walk between u and v} be the toll interval between u and v in G.In the associated toll convexity, a subset Next, we introduce the concept of weakly toll convexity.Let u, v ∈ V (G).A weakly toll walk between u and v in G is a sequence of vertices of the form In other words, a weakly toll walk is any walk W : u, w 1 , . . ., w k−1 , v between u and v such that u is adjacent only to the vertex w 1 , which can appear more than once in the walk, and v is adjacent only to the vertex w k−1 , which can appear more than once in the walk.Note that if uv ∈ E(G), then W : u, v is a weakly toll walk, and if k = 0 then W : u is a wealky toll walk consisting of a single vertex.We define WT G (u, v) = {x ∈ V (G) : x lies in a weakly toll walk between u and v} to be the weakly toll interval between u and v in G. Finally, a subset Note that any weakly toll convex set is also a toll convex set.Also, any toll convex set is a monophonically convex set, and a monophonically convex set is a geodesically convex set.
On the other hand, consider the graph K 1,3 with vertices a, b, c, d, where b is the vertex with degree three, and let S = {a, b, c}.It is clear that S is toll convex but not weakly toll convex, since a, b, d, b, c is a weakly toll walk between a and c that contains d / ∈ S.
The weakly toll convex hull of a set S ⊆ V (G) is the smallest set of vertices in G that contains S and is weakly toll convex (alternatively, it is the intersection of all weakly toll convex sets that contain S).In this particular convexity, the concept of extreme vertex can also be defined.A vertex x of a weakly toll convex set S of a graph G is an extreme vertex of S if S \ {x} is also a weakly toll convex set in G.
A graph is an interval graph if it has an intersection model (or interval model) consisting of closed intervals on a straight line.Given an interval I, let R(I) and L(I) be, respectively, the right and left endpoints of I. Given a family of intervals {I v } v∈V (G) , we say that I a is an end interval if L(I a ) = Min I v or R(I a ) = Max I v .A given vertex a in an interval graph G is an end vertex if there exists some interval model where a is represented by an end interval.
A graph is chordal if every cycle of length at least four has a chord.
Theorem 2.1.(Dirac, 1961) A graph is chordal if and only if it has a perfect elimination order.Lekkerkerker and Boland (1962) proved that a chordal graph is an interval graph if and only if it contains no asteroidal triple.Three vertices of a graph form an asteroidal triple if for any two of them, there exists a path between them that does not intersect the closed neighborhood of the third.Gimbel (1988) studied the end vertices of an interval graph: Theorem 2.2.(Gimbel, 1988) A vertex v of an interval graph G is an end vertex if and only if G does not contain any of the graphs in Figure 1 as an induced subgraph with v as the designated vertex.
A simplicial vertex v of an interval graph G is called end simplicial vertex if it is an end vertex of G.
In Figure 1, consider the graphs star 1,2,2 , B n (n > 5), and the bull graph.Note that v is a simplicial vertex in any of such graphs.In addition, consider the walks (in the bull graph).Note that these walks are weakly toll walks.Hence v is not an extreme vertex of any graph in Figure 1.
A proper interval graph is an interval graph that has a proper interval model, i.e., an interval model in which no interval strictly contains another.More precisely, there exists a collection of intervals Roberts (1969) proved that proper interval graphs are exactly the K 1,3 -free interval graphs.Observe that star 1,2,2 and B n (see Figure 1) are not proper interval graphs.In addition, in a proper interval model of the graph P 5 , the end intervals are necessarily associated with the vertices of degree one.Therefore, we have the following corollary: Corollary 2.1.A vertex v of a proper interval graph G is an end vertex if and only if G does not contain any of the graphs in Figure 2 as an induced subgraph with v as the designated vertex.In order to determine whether a graph is a convex geometry with respect to some convexity, it is important to know which vertices are the extreme vertices of convex sets in that convexity.As said above, chordal graphs are associated with the monophonic convexity, and in this case the extreme vertices are exactly the simplicial vertices (Farber and Jamison, 1986).In the case of interval graphs, associated with the toll convexity, the extreme vertices are the end simplicial vertices (Alcón, Brešar, Gologranc, Gutierrez, Šumenjak, Peterin, and Tepeh, 2015).
Theorem 2.3.(Alcón, Brešar, Gologranc, Gutierrez, Šumenjak, Peterin, and Tepeh, 2015) A graph is a convex geometry with respect to the toll convexity if and only if it is an interval graph.
Lemma 2.1.Let G be an interval graph.Every vertex of G which is not an end simplicial vertex lies in a weakly toll walk between two end simplicial vertices.
Proof: Since G is an interval graph, by Theorem 2.3, G is a convex geometry with respect to the toll convexity.This means that every vertex of G that is not an end simplicial vertex lies in a tolled walk between two end simplicial vertices.As every tolled walk is a weakly toll walk, the lemma follows.
Lemma 2.2.Let C be a weakly toll convex set of a graph G.If Then there exist two neighbors of x in C, say u and v, which are not adjacent.But then u, x, v is a weakly toll walk in G.
Hence C \ {x} is not a weakly toll convex set, which is the desired contradiction.

Lemma 2.3. A vertex v of a proper interval graph G is an extreme vertex of the weakly toll convex set V (G) if and only if v is an end simplicial vertex of G.
Proof: Suppose that v is an extreme vertex that is not an end simplicial vertex.Since v is an extreme vertex, no induced path contains v as an internal vertex; hence, v is a simplicial vertex so that v is not an end vertex.Using Corollary 2.1 and the observations after Theorem 2.2, v is not an extreme vertex, which is a contradiction.

M.C. Dourado, M. Gutierrez, F. Protti, S. Tondato
Conversely, assume, in order to obtain a contradiction, that there exists a weakly toll walk W between two nonadjacent vertices x and y of G containing an end simplicial vertex v as an internal vertex.Write W : x, w 1 , . . ., w i , v, w i+2 , . . ., w n , y.Since v is an end simplicial vertex, we can assume that there exists a proper interval model {I u } u∈V (G) such that I v appears as the first interval on the line.First, we will show that x is not adjacent to v. Suppose that x is adjacent to v.As W is a weakly toll walk, w 1 = v.Moreover, since v is a simplicial vertex, w i and w i+2 are adjacent to x.Thus w i = w i+2 = v.
Clearly w k = v for k ∈ {1, . . ., n}.Then W is the walk x, v, y.Since v is a simplicial vertex and x, y are adjacent to v, we have that x is adjacent to y, which is a contradiction.
Since x and y are not adjacent to v, and x is not adjacent to y, we have I v ∩ I x = ∅, I x ∩ I y = ∅, and I v ∩ I y = ∅.Thus, we can assume that I v , I x , and I y appear in this order on the line.
On the other hand, the weakly toll walk W : x, w 1 , . . ., w i , v, w i+2 , . . ., w n , y goes from x to v through w 1 , w 1 is the only vertex adjacent to x, and G is a proper interval graph.Thus L(I x ) ∈ I w1 and L(I w1 ) < L(I x ).Also, the weakly toll walk W : x, w 1 , . . ., w i , v, w i+2 , . . ., w n , y goes from v to y through w 1 because the only vertex adjacent to x is w 1 .Thus R(I x ) ∈ I w1 and R(I x ) < R(I w1 ).
Hence, we have Lemma 2.4.
1. Let x, y and z be vertices forming an asteroidal triple in a graph G.If C is the weakly toll convex hull of the set {x, y, z}, then C does not have extreme vertices.
2. Let a, b, c, and d be vertices inducing a K 1,3 in a graph G.If C is the weakly toll convex hull of the set {a, b, c, d}, then C does not have extreme vertices.

Proof:
1. Let w be a vertex of C, different from x, y, and z.Assume, in order to obtain a contradiction, that w is an extreme vertex in C. Thus C \ {w} is a weakly toll convex set.Also C \ {w} contains x, y, and z.This is a contradiction.Now, we will show that no vertex of the set {x, y, z} is an extreme vertex.Let P and Q be the induced paths between x, y (avoiding neighbors of z), and y, z (avoiding neighbors of x), respectively.By concatenating P and Q, we obtain a weakly toll walk between x and z containing y (observe that no vertex of Q can be adjacent to x, and no vertex of P can be adjacent to z).Thus, there is a weakly toll walk between two vertices of the asteroidal triple containing the other vertex of the asteroidal triple.Hence no vertex forming the asteroidal triple is an extreme vertex in C. Weakly toll convexity and proper interval graphs 7

As in the
3 Proper interval graphs as convex geometries Recall that if a graph is a convex geometry with respect to a particular convexity, then V (G) and every convex subset of G is the convex hull of its extreme vertices with respect to that convexity.
Theorem 3.1.A graph G is a convex geometry with respect to the weakly toll convexity if and only if G is a proper interval graph.
Proof: Let G be a convex geometry w.r.t. the weakly toll convexity.We will show that G is a proper interval graph, using induction on the number of vertices of G.The claim is true if G has one or two vertices.Assume that G has n vertices and that the claim is true for all graphs with fewer than n vertices.Let x be any extreme vertex of V (G).Clearly G − x is a convex geometry.By the induction hypothesis, G − x is a proper interval graph.In particular, G − x is a chordal graph.By Lemma 2.2, x is a simplicial vertex in G; thus, G is chordal.If G has an asteroidal triple, then, by Lemma 2.4, the weakly toll convex hull C of the asteroidal triple has no extreme vertices.This implies that C is not the weakly toll convex hull of its extreme vertices, which is a contradiction.Thus G is a chordal graph with no asteroidal triple, and this implies that it is an interval graph.Now, we claim that G does not contain K 1,3 as an induced subgraph.In order to obtain a contradiction, suppose that G contains G ′ = K 1,3 as an induced subgraph.
Let C be the weakly toll convex hull of the set V (G ′ ).It is clear, using Lemma 2.4, that C has no extreme vertices.This implies that C is not the weakly toll convex hull of its extreme vertices, which is a contradiction.Hence, G is a proper interval graph.
Conversely, every convex subset of a proper interval graph G induces a proper interval graph.Thus it suffices to show that V (G) is the convex hull of its extreme vertices.Since G is a proper interval graph, then it is an interval graph.By Lemma 2.1, every vertex of G that is not an end simplicial vertex lies in a wealky toll walk between two end simplicial vertices.Recall that, by Lemma 2.3, the set of end simplicial vertices is equal to the set of extreme vertices in the weakly toll convexity if G is a proper interval graph.
Hence every vertex of G that is not an extreme vertex lies in a weakly toll walk between two extreme vertices.This implies that G is a convex geometry.
4 Some invariants associated with the weakly toll convexity In this section, we consider some standard invariants with respect to the weakly toll convexity that have been extensively studied for other graph convexities.We consider the weakly toll number and the weakly toll hull number of a graph.
The definition of weakly toll interval for two vertices u and v can be generalized to an arbitrary subset S of V (G) as follows: The order of a minimum weakly toll set in G is called the weakly toll number of G, and is denoted by wtn(G).For any non-trivial connected graph G, it is clear that 2 ≤ wtn(G) ≤ n.
As mentioned earlier, the weakly toll convex hull of a set S ⊆ V (G) is defined as the intersection of all weakly toll convex sets that contain S, and we will denote this set by [S] WT .A set S is a weakly toll The weakly toll hull number of G, denoted by wth(G), is the minimum among all the cardinalities of weakly toll hull sets.
Given a set S ⊆ V (G), define WT k (S) as follows: WT 0 (S) = S and From the definitions, we immediately infer that every weakly toll set is a weakly toll hull set, and hence wth(G) ≤ wtn(G).
By the proof of Theorem 3.1, the weakly toll number (as well as the weakly toll hull number) of a proper interval graph coincides with the number of its extreme vertices.Indeed, every weakly toll set of G contains Ext(G).Furthermore, every vertex that is not an extreme vertex lies in a weakly toll interval between two extreme vertices.We then derive the following fact: It is known that, among the trees, only paths are proper interval graphs.In the following theorem, we determine the weakly toll number and the weakly toll hull number of any tree.
Proof: Let a and b be two leaves of G. Clearly, every vertex of V (G)\{a, b} lies in some weakly toll walk between a and b; hence wtn(G) ≤ 2. Since G is a non-trivial graph, wtn(G) = 2.As wth(G) ≤ wtn(G) and G is a non-trivial graph, it follows that wth(G) = 2.
We now need to recall a special representation of interval graphs.More details can be found in (Fulkerson and Gross, 1965).Let C(G) be the set of all maximal cliques of an interval graph G.A canonical representation I of G is a total order Q 1 , . . ., Q k of the set C(G) in which for each vertex v of G, the cliques in the set Q v = {Q ∈ C(G) | v ∈ Q} occur consecutively in the order (see Figure 3).Maximal cliques Q 1 and Q k are called end cliques of the representation.In addition, for two maximal cliques and there is an edge We denote by G I the acyclic spanning subgraph of K(G) consisting of the path Q 1 , Q 2 , . . ., Q k .The graph G I is intrinsically associated with the canonical representation I.It is a particular case of a clique tree of a chordal graph (recall that interval graphs are chordal).For more details, see (Buneman, 1974;Gavril, 1974).In the case of interval graphs, clique trees are paths.Note that in every canonical representation I of star 1,2,2 , B n (n > 5), and the bull graph, N [v] is a clique that is not an end clique (see Figure 1).Moreover, if G is an interval graph which contains star 1,2,2 , B n (n > 5), or the bull graph as an induced subgraph and In this context, a given simplicial vertex v in an interval graph G is an end vertex if there exists some canonical representation where N [v] is an end clique of this representation.This can be proved as follows: if v is a simplicial vertex and N [v] is an end clique of a canonical representation, then v cannot be the designated vertex in any configuration depicted in Figure 1.Thus, by Theorem 2.2, v is an end vertex.
In what follows, we show some properties of the canonical representation I of an interval graph G. Proposition 4.2.Let G be a connected interval graph, and If the vertices appear in the order and the proof uses the same arguments.
Proposition 4.3.Let G be a connected interval graph, I a canonical representation of G, and Proposition 4.4.Let G be a connected interval graph, I a canonical representation of G, and s an end simplicial vertex of G such that N [s] and Q are the end cliques of I.
connected components, then there exist k vertices q 1 , . . ., q k in G − N [s] such that: 1. every q i is an end simplicial vertex of G; appear in this order or in reverse order in I; 4. for i = 1, . . ., k, there exists a maximal clique and 5. for i = 1, . . ., k, there exists another canonical representation I i such that N [q i ] and Q are its end cliques; , and the vertices of has exactly two connected components.In order to fix ideas, assume that is clearly associated with another canonical representation I i of G, for i = 1, . . ., k.Note that Q i is an end clique of I i , so there exist simplicial end vertices q i ∈ Q i , which are end vertices of G, for i = 1, . . ., k.As q i is simplicial, N [q i ] is a maximal clique of G and Proposition 4.5.Let G be a connected interval graph, I a canonical representation of G, and s 1 and ) lies in a weakly toll walk between s 1 and s 2 .
Proof: Let y and w be vertices of G such that y ∈ N (s 1 ), w ∈ N (s 2 ), and |Q v | is maximum for each v ∈ {y, w}.Note that y may be equal to w.Let x be a vertex of If y = w or y, w are twins then Q y = C(G).Therefore, W : s 1 , y, x, y, s 2 is a weakly toll walk between s 1 and s 2 , which captures x.
Assume that y = w and y, w are not twins.If y is adjacent to w, then there is a maximal clique Q ∈ C(G) containing both y and w.Thus, appear in this order (or in reverse order) in I. Suppose without loss os generality that N [s 1 ], Q, N [s 2 ] appear in this order in I. Since the cliques in Q y (or in Q w ) occur consecutively in I, Q y contains all the cliques in I from N [s 1 ] to Q, and Q w contains all the cliques in . Moreover, x is clearly adjacent to y or w.Thus either W 1 : s 1 , y, x, y, w, s 2 , W 2 : s 1 , y, w, x, w, s 2 , or W 3 : s 1 , y, x, w, s 2 is a weakly toll walk between s 1 and s 2 , which captures x.Now, suppose that y is not adjacent to w.Let P : y, x 1 , .., x n , w be an induced path between y and w in G.By the choice of y and w, Let x 0 = y and x n+1 = w.Since x i x i+1 ∈ E(G), let Q i be a clique of C(G) containing both x i and x i+1 , for i = 0, . . ., n.Since P is induced, Q i ∩ {x 0 , . . ., x n+1 } = {x i , x i+1 }, and this implies that (a) appear in this order (or in reverse order) in I. Suppose without loss of generality that To conclude the proof, observe that x = x i or x must be adjacent to y or w or x i for some 1 ≤ i ≤ n.Therefore, s 1 , P, s 2 or s 1 , y, x, P, s 2 or s 1 , P, x, w, s 2 or s 1 , y, x 1 , . . ., x i , x, x i , . . ., x n , w, s 2 (for some i ∈ {1, . . ., n}) is a weakly toll walk between s 1 and s 2 , which captures x.
Proposition 4.6.Let G be a connected interval graph, let x is a twin of s i } for i = 1, 2. Then, (1) S 1 ∪ S 2 is a weakly toll set of G; (2) if there exists v / ∈ N [s 1 ] such that for some y ∈ N [s 1 ], Q v Q y , and there exists an induced path P between y and s 2 with P ∩ N [v] = {y}, then S = S 1 ∪ {s 2 , v} is a weakly toll set of G; (3) if there exist v / ∈ N [s 1 ] and w / ∈ N [s 2 ] such that for some y ∈ N [s 1 ] and z ∈ N [s 2 ], Q v Q y and Q w Q z , and there exist induced paths P between y and s 2 with P ∩ N [v] = {y} and P 1 between z and s 1 with P 1 ∩ N [w] = {z}, then S = {s 1 , v, s 2 , w} is a weakly toll set of G; Proof: . By Proposition 4.5, x lies in a weakly toll walk between s 1 and s 2 .Now, for i ∈ {1, 2}, let x ∈ N [s i ] \ S i .Since x / ∈ S i and G − N [s i ] is a connected graph, there exists at least a vertex y ∈ N (x) \ N [s i ] and an induced path P : y, . . ., s j avoiding the neighbors of s i .Note that no vertex of P is a neighbor of s i .Then s i , x, y, . . ., s j is a walk between s i and s j which contains x as internal vertex; in addition, this walk contains an induced path P 1 between s i and s j which contains x as an internal vertex.Hence P 1 is a weakly toll walk between s i and s j containing x as an internal vertex.Thus S 1 ∪ S 2 is a weakly toll set of G and wtn(G) ≤ |S 1 ∪ S 2 |.
(3) Every vertex of V (G) \ (S 1 ∪ S 2 ) is contained in the weakly toll interval of {s 1 , s 2 }, every vertex of S 2 is contained in the weakly toll interval of {s 1 , v}, and every vertex of S 1 is contained in the weakly toll interval of {s 2 , w}.Hence S = {s 1 , v, s 2 , w} is a weakly toll set and wtn(G) ≤ |S|.
By the above proposition, the weakly toll number of an interval graph having two simplicial vertices s 1 and s 2 , which are always in end cliques of every representation, is a number between two and the number of twins of s 1 and s 2 plus two.The following example is elucidative.In Figure 4, the graph has weakly toll number three.Observe that there is a weakly toll walk W between 1 and 4, namely W : 1, 3, 4, which captures vertex 3, but there is no weakly toll walk between 1 and 4 which captures vertex 2. Hence, {1, 2, 4} is a weakly toll set of G.
In the following theorem, we show that an interval graph with no end simplicial vertex having a twin has weakly toll number at most three and weakly toll hull number equal to two.Proof: First, we will prove that wtn(G) ≤ 3.
Let s 2 be an end vertex of G such that N [s 2 ] is the other end clique of the canonical representation I.
] lies in a weakly toll walk between s 1 and s 2 .
On the other hand, if x ∈ N [s 1 ] ∩ N [s 2 ] then x lies in the weakly toll walk W = s 1 , x, s 2 , between s 1 and s 2 .
Thus we just need to study vertices

Weakly toll convexity and proper interval graphs 13
Assume that G − N [s 1 ] has at least two connected components.By Proposition 4.4, if G − N [s 1 ] has k + 1 connected components, there exist end simplicial vertices q 1 , . . ., q k of G and Q Fig. 5: Let w 1 and w n be vertices in N (s 1 ) and N (s 2 ), respectively, such that |Q w1 | and |Q wn | are maximum.Clearly, q 1 ∈ N (w 1 ).
First, suppose that x ∈ N (s 1 ).If x ∈ N (q 1 ) then s 1 , x, q 1 is a weakly toll walk.
Assume that x / ∈ N (q 1 ).If w 1 and w n are twins, then s 2 , w 1 , x, w 1 , q 1 is a weakly toll walk.
Assume now that w 1 and w n are not twins.Note that q is not a connected graph; in addition, there exist vertices a ∈ N (w 1 ) ∩ V (C) and b ∈ N (w n ) ∩ V (C).Let a, y 1 , . . ., y m , b be an induced path between a and b.Let i and j be the first and the last indices such that y i is adjacent to w n and y j is adjacent to w 1 .Since w 1 is not adjacent to w n , j ≤ i.Thus, the walk q 1 , w 1 , x, w 1 , y j , y j+1 , . . ., y i , w n , s 2 is a weakly toll walk.Now, suppose that x ∈ N (s 2 ).If w 1 and w n are twins, then s 1 , w n , x, w n , q 1 is a weakly toll walk.Consider now the case in which w 1 and w n are not twins.In this situation, q 1 / ∈ N (w n ).If Q w1 ∩Q wn = ∅ then s 1 , w 1 , w n , x, w n , w 1 , q 1 is a weakly toll walk.
Suppose now that Q w1 ∩ Q wn = ∅.Thus, as exposed above, there exists a connected component C of G \ (N [s 1 ] ∪ N [s 2 ]) whose vertices are in G[Q ′ k , Q]; in addition, there exist vertices a ∈ N (w 1 ) ∩ V (C) and b ∈ N (w n ) ∩ V (C).Let a, y ′ 1 , . . ., y ′ p , b an induced path between a and b.Let i and j be the first and the last indices such that y ′ i is adjacent to w n and y ′ j is adjacent to w 1 .Since w 1 is not adjacent to w n , j ≤ i.Thus, the walk q 1 , y ′ j , . . ., y ′ i , w n , x, w n , y ′ i , . . ., y ′ j , w 1 , s 1 is a weakly toll walk.From the above, S = {s 1 , s 2 , q 1 } is a weakly toll set of G. Figures 5, 6 illustrate the idea of the proof.Therefore, wtn(G) ≤ 3.
In order to prove that wth(G) = 2, first observe that, by Proposition 4.5, every end simplicial vertex, Weakly toll convexity and proper interval graphs 15 distinct from s 1 and s 2 , lies in a weakly toll walk between s 1 and s 2 .
Let x be a vertex of G that is not an end simplicial vertex.Since G is an interval graph, every vertex x of G that is not an end simplicial vertex lies in a tolled walk between two end simplicial vertices, and every end simplicial vertex lies in a weakly toll walk between s 1 and s 2 ; thus, it is possible to build a weakly toll walk between s 1 and s 2 which captures x using vertices of both walks.
In the remainder of this section, we consider an interval graph G, and study the weakly toll number of G. Let s 1 and s 2 be two end simplicial vertices of G such that N Suppose that G − N [s i ] is not a connected graph for some i ∈ {1, 2}.By Theorem 4.2, S = {s 1 , q 1 , s 2 } is a weakly toll set of G.
From the above, we obtain the following result.

Conclusions
In this work, we introduced a new graph convexity based on the concept of weakly toll walks, which generalize induced paths, and showed how such a convexity gives rise to a new structural characterization of proper interval graphs.Also, we found bounds for the weakly toll number and the weakly toll hull number of an arbitrary interval graph.
We propose a further study of these two invariants in general graphs, namely, determining the graphs G for which wtn(G) = wth(G) = |Ext(G)|.Other research direction is the study of the Carathéodory number, the Radon number, and the Helly number in the context of the weakly toll convexity.Finally, characterizing weakly toll convex sets in graph products is also an interesting open problem.

Fig. 2 :
Fig. 2: Forbidden configurations for an end vertex in proper interval graphs.
from a, b, c, and d, then w is not an extreme vertex.Let a, b, c, d be the vertices of a K 1,3 such that b has degree three.Since a, b, c, b, d is a weakly toll walk between a and d, b and c are not extreme vertices of K 1,3 .Likewise, we prove that a and d are not extreme vertices of K 1,3 .
Theorem 4.2.Let G be a connected interval graph.Let s 1 be an end vertex ofG such that N [s 1 ] is an end clique of a canonical representation I of G.If G − N [s 1 ]is not a connected graph, then wtn(G) ≤ 3 and wth(G) = 2.

Fig. 6 :
Fig. 6: The walk s1, d, b, q1, d, c, d, e, q ′1 , e, s2 is a weakly toll walk which captures V (G) \ {a, f }.The walk s1, a, b, q1 is a weakly toll walk which captures a and b.Finally, the walk s1, d, e, f, e, d, q1 is a weakly toll walk which captures f .
[s 1 ] and N [s 2 ] are the end cliques of a canonical representation I of G.If G − N [s i ] is a connected graph for i = 1, 2, then, by Proposition 4.6, it follows that wtn(G) ≤ |S 1 ∪ S 2 |.
Corollary 4.1.Let G be an interval graph, s 1 , s 2 be two end simplicial vertices of G, which are not twins, such that N [s 1 ] and N [s 2 ] are end cliques of a canonical representation I of G, andS i = {s i } ∪ {x ∈ V (G) : x is a twin of s i } for i = 1, 2. Then, 2 ≤ wtn(G) ≤ |S 1 ∪ S 2 |.