47,151 research outputs found
Graphs with four boundary vertices
18 pagesInternational audienceA vertex v of a graph G is a boundary vertex if there exists a vertex u such that the distance in G from u to v is at least the distance from u to any neighbour of v. We give a full description of all graphs that have exactly four boundary vertices, which answers a question of Hasegawa and Saito. To this end, we introduce the concept of frame of a graph. It allows us to construct, for every positive integer b and every possible ``distance-vector'' between b points, a graph G with exactly b boundary vertices such that every graph with b boundary vertices and the same distance-vector between them is an induced subgraph of G
A characterization of graphs with at most four boundary vertices
Steinerberger defined a notion of boundary for a graph and established a
corresponding isoperimetric inquality. Hence, "large" graphs have more boundary
vertices. In this paper, we first characterize graphs with three boundary
vertices in terms of two infinite families of graphs. We then completely
characterize graphs with four boundary vertices in terms of eight families of
graphs, five of which are infinite. This parallels earlier work by Hasegawa and
Saito as well as M\"uller, P\'or, and Sereni on another notion of boundary
defined by Chartrand, Erwin, Johns, and Zhang.Comment: 16 pages, 9 figure
4-colored graphs and knot/link complements
A representation for compact 3-manifolds with non-empty non-spherical
boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper
edge-coloration with four colors) has been recently introduced by two of the
authors, and an initial classification of such manifolds has been obtained up
to 8 vertices of the representing graphs. Computer experiments show that the
number of graphs/manifolds grows very quickly as the number of vertices
increases. As a consequence, we have focused on the case of orientable
3-manifolds with toric boundary, which contains the important case of
complements of knots and links in the 3-sphere. In this paper we obtain the
complete catalogation/classification of these 3-manifolds up to 12 vertices of
the associated graphs, showing the diagrams of the involved knots and links.
For the particular case of complements of knots, the research has been extended
up to 16 vertices.Comment: 19 pages, 6 figures, 3 tables; changes in Lemma 6, Corollaries 7 and
Genus Ranges of 4-Regular Rigid Vertex Graphs
We introduce a notion of genus range as a set of values of genera over all
surfaces into which a graph is embedded cellularly, and we study the genus
ranges of a special family of four-regular graphs with rigid vertices that has
been used in modeling homologous DNA recombination. We show that the genus
ranges are sets of consecutive integers. For any positive integer , there
are graphs with vertices that have genus range for all
, and there are graphs with vertices with genus range
for all or . Further, we show that
for every there is such that is a genus range for graphs with
and vertices for all . It is also shown that for every ,
there is a graph with vertices with genus range , but there
is no such a graph with vertices
Extremal fullerene graphs with the maximum Clar number
A fullerene graph is a cubic 3-connected plane graph with (exactly 12)
pentagonal faces and hexagonal faces. Let be a fullerene graph with
vertices. A set of mutually disjoint hexagons of is a sextet
pattern if has a perfect matching which alternates on and off each
hexagon in . The maximum cardinality of sextet patterns of is
the Clar number of . It was shown that the Clar number is no more than
. Many fullerenes with experimental evidence
attain the upper bound, for instance, and . In
this paper, we characterize extremal fullerene graphs whose Clar numbers equal
. By the characterization, we show that there are precisely 18
fullerene graphs with 60 vertices, including , achieving the
maximum Clar number 8 and we construct all these extremal fullerene graphs.Comment: 35 pages, 43 figure
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
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