A toll walk W= w 1⋯ wk in a graph G is a walk in which w 1 is adjacent only to w 2 and wk is
adjacent only to wk-1 among all vertices of W. The toll walk interval T (u, v) between u, v∈ V …

Graph convexity has been used as an important tool to better understand the structure of
classes of graphs. Many studies are devoted to determine if a graph equipped with a …

Toll walks on connected graphs are introduced to characterize dominating pairs of vertices
in interval graphs. A weak-toll walk is an immediate generalization of a toll walk in a graph …

A tolled walk T between two non-adjacent vertices u and v in a graph G is a walk, in which u
is adjacent only to the second vertex of T and v is adjacent only to the second-to-last vertex …

Toll convexity is a variation of the so-called interval convexity. A tolled walk T between two
non-adjacent vertices u and v in a graph G is a walk, in which u is adjacent only to the …

Let $ V $ be a finite nonempty set. A transit function is a map $ R: V\times V\rightarrow 2^ V $
such that $ R (u, u)=\{u\} $, $ R (u, v)= R (v, u) $ and $ u\in R (u, v) $ hold for every $ u, v\in V …

In this paper we study domination between different types of walks connecting two non-
adjacent vertices of a graph. In particular, we center our attention on weakly toll walk and lk …

The exploration of weakly toll convexity is the focus of this investigation. A weakly toll walk is
any walk $ W: u, w_1,\ldots, w_ {k-1}, v $ between $ u $ and $ v $ such that $ u $ is adjacent …

A walk $ W= w_1w_2\dots w_k $, $ k\geq 2$, is called a toll walk if $ w_1\neq w_k $ and $
w_2 $ and $ w_ {k-1} $ are the only neighbors of $ w_1 $ and $ w_k $, respectively, on $ W …

HHD-free is the class of graphs which contain no house, hole, or domino as induced
subgraph. It is known that HHD-free graphs can be characterized via LexBFS-ordering and …