Weakly toll convexity and proper interval graphs

Mitre C. Dourado, Marisa Gutierrez, Fábio Protti, Silvia B. Tondato

Abstract: A walk

u_{0} u_{1} l d o t s u_{k - 1} u_{k}

is a extit{weakly toll walk} if

u_{0} u_{i} i n E (G)

implies

u_{i} = u_{1}

and

u_{j} u_{k} i n 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 i n 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

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.

Mathematics Subject Classification ID

Structural characterization of families of graphs (05C75)

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