1,196 research outputs found
Independent [1,2]-number versus independent domination number
A [1, 2]-set S in a graph G is a vertex subset such that every vertex
not in S has at least one and at most two neighbors in it. If the additional
requirement that the set be independent is added, the existence of such
sets is not guaranteed in every graph. In this paper we provide local
conditions, depending on the degree of vertices, for the existence of
independent [1, 2]-sets in caterpillars. We also study the relationship
between independent [1, 2]-sets and independent dominating sets in this
graph class, that allows us to obtain an upper bound for the associated
parameter, the independent [1, 2]-number, in terms of the independent
domination number.Peer ReviewedPostprint (published version
On the excessive [m]-index of a tree
The excessive [m]-index of a graph G is the minimum number of matchings of
size m needed to cover the edge-set of G. We call a graph G [m]-coverable if
its excessive [m]-index is finite. Obviously the excessive [1]-index is |E(G)|
for all graphs and it is an easy task the computation of the excessive
[2]-index for a [2]-coverable graph. The case m=3 is completely solved by
Cariolaro and Fu in 2009. In this paper we prove a general formula to compute
the excessive [4]-index of a tree and we conjecture a possible generalization
for any value of m. Furthermore, we prove that such a formula does not work for
the excessive [4]-index of an arbitrary graph.Comment: 12 pages, 7 figures, to appear in Discrete Applied Mathematic
On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids
The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest
integer k such that for every colouring of the vertices of H with exactly k
colours, there is a hyperedge of H all of whose vertices have different
colours. We denote by nu(H) the number of vertices of H and by tau(H) the size
of the smallest set containing at least two vertices of each hyperedge of H.
For a complete geometric graph G with n > 2 vertices let H = H(G) be the
hypergraph whose vertices are the edges of G and whose hyperedges are the edge
sets of plane spanning trees of G. We prove that if G has at most one interior
vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) -
tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given
by the ground set and the bases of a matroid, respectively
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
Graph classes and forbidden patterns on three vertices
This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page
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