14 research outputs found
Types of triangle in plane Hamiltonian triangulations and applications to domination and k-walks
We investigate the minimum number t(0)(G) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t(0)(G) faces that do not contain an edge of C. We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3-cuts are constructed from smaller Hamiltonian cycles of 4-connected subgraphs. We also present results linking the number of these triangles to the length of 3-walks in a class of triangulation and to the domination number
Domination number of annulus triangulations
An {\em annulus triangulation} is a 2-connected plane graph with two disjoint faces and such that every face other than and are triangular, and that every vertex of is contained in the boundary cycle of or . In this paper, we prove that every annulus triangulation with vertices of degree 2 has a dominating set with cardinality at most if is not isomorphic to the octahedron. In particular, this bound is best possible
Isolation of squares in graphs
Given a set of graphs, we call a copy of a graph in
an -graph. The -isolation number of a
graph , denoted by , is the size of a smallest subset
of the vertex set such that the closed neighbourhood of
intersects the vertex sets of the -graphs contained by
(equivalently, contains no -graph). Thus,
is the domination number of . The second author showed
that if is the set of cycles and is a connected -vertex
graph that is not a triangle, then . This bound is attainable for every and solved
a problem of Caro and Hansberg. A question that arises immediately is how
smaller an upper bound can be if for some ,
where is a cycle of length . The problem is to determine the smallest
real number (if it exists) such that for some finite set
of graphs, for every connected graph
that is not an -graph. The above-mentioned result yields and . The second author also showed that
if and exists, then . We prove that
and determine , which consists of three
-vertex graphs and six -vertex graphs. The -vertex graphs in
were fully determined by means of a computer program. A method
that has the potential of yielding similar results is introduced.Comment: 15 pages, 1 figure. arXiv admin note: text overlap with
arXiv:2110.0377
A generalization of the Art Gallery Theorem
Several domination results have been obtained for maximal outerplanar graphs
(mops). The classical domination problem is to minimize the size of a set
of vertices of an -vertex graph such that , the graph obtained
by deleting the closed neighborhood of , contains no vertices. A classical
result of Chv\'{a}tal, the Art Gallery Theorem, tells us that the minimum size
is at most if is a mop. Here we consider a modification by allowing
to have a maximum degree of at most . Let denote the
size of a smallest set for which this is achieved. If , then
trivially . Let be a mop on
vertices, of which are of degree~. Sharp bounds on have
been obtained for and , namely and . We prove that
for any , and that this bound is sharp. We also prove that
is a sharp upper bound on the domination number of .Comment: arXiv admin note: substantial text overlap with arXiv:1903.1229