63 research outputs found
Split graphs and Block Representations
In this paper, we study split graphs and related classes of graphs from the
perspective of their sequence of vertex degrees and an associated lattice under
majorization. Following the work of Merris in 2003, we define blocks
, where is the degree sequence of a graph, and
and are sequences arising from . We use the
block representation to characterize membership in
each of the following classes: unbalanced split graphs, balanced split graphs,
pseudo-split graphs, and three kinds of Nordhaus-Gaddum graphs (defined by
Collins and Trenk in 2013). As in Merris' work, we form a poset under the
relation majorization in which the elements are the blocks
representing split graphs with a fixed number of
edges. We partition this poset in several interesting ways using what we call
amphoras, and prove upward and downward closure results for blocks arising from
different families of graphs. Finally, we show that the poset becomes a lattice
when a maximum and minimum element are added, and we prove properties of the
meet and join of two blocks.Comment: 23 pages, 7 Figures, 2 Table
On the algorithmic complexity of twelve covering and independence parameters of graphs
The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs
Bounds for the Number of Independent and Dominating Sets in Trees
In this work, we investigate bounds on the number of independent sets in a graph and its complement, along with the corresponding question for number of dominating sets. Nordhaus and Gaddum gave bounds on χ(G)+χ(G) and χ(G) χ(G), where G is any graph on n vertices and χ(G) is the chromatic number of G. Nordhaus-Gaddum- type inequalities have been studied for many other graph invariants. In this work, we concentrate on i(G), the number of independent sets in G, and ∂(G), the number of dominating sets in G. We focus our attention on Nordhaus-Gaddum-type inequalities over trees on a fixed number of vertices. In particular, we give sharp upper and lower bounds on i(T )+ i(T ) where T is a tree on n vertices, improving bounds and proofs of Hu and Wei. We also give upper and lower bounds on i(G) + i(G) where G is a unicyclic graph on n vertices, again improving a result of Hu and Wei. Lastly, we investigate ∂(T )+ ∂(T ) where T is a tree on n vertices. We use a result of Wagner to give a lower bound and make a conjecture about an upper bound
Transitivity on subclasses of chordal graphs
Let be a graph, where and are the vertex and edge sets,
respectively. For two disjoint subsets and of , we say
\textit{dominates} if every vertex of is adjacent to at least one
vertex of in . A vertex partition of
is called a \emph{transitive -partition} if dominates for
all , where . The maximum integer for which the above
partition exists is called \emph{transitivity} of and it is denoted by
. The \textsc{Maximum Transitivity Problem} is to find a transitive
partition of a given graph with the maximum number of partitions. It was known
that the decision version of \textsc{Maximum Transitivity Problem} is
NP-complete for chordal graphs [Iterated colorings of graphs, \emph{Discrete
Mathematics}, 278, 2004]. In this paper, we first prove that this problem can
be solved in linear time for \emph{split graphs} and for the \emph{complement
of bipartite chain graphs}, two subclasses of chordal graphs. We also discuss
Nordhaus-Gaddum type relations for transitivity and provide counterexamples for
an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The
transitivity of a graph, \emph{J. Combin. Math. Combin. Comput}, 104, 2018].
Finally, we characterize transitively critical graphs having fixed
transitivity.Comment: arXiv admin note: text overlap with arXiv:2204.1314
Clique coverings and claw-free graphs
Let C be a clique covering for E(G) and let v be a vertex of G. The valency of vertex v (with respect to C), denoted by val(C) (v), is the number of cliques in C containing v. The local clique cover number of G, denoted by lcc(G), is defined as the smallest integer k, for which there exists a clique covering for E(G) such that val(C) (v) is at most k, for every vertex v is an element of V(G). In this paper, among other results, we prove that if G is a claw-free graph, then lcc(G) + chi(G) <= n + 1. (C) 2020 The Author(s). Published by Elsevier Ltd
Characterizations for split graphs and unbalanced split graphs
We introduce a characterization for split graphs by using edge contraction.
Then, we use it to prove that any (, claw)-free graph with is a split graph. Also, we apply it to characterize any pseudo-split
graph. Finally, by using edge contraction again, we characterize unbalanced
split graphs which we use to characterize the Nordhaus-Gaddum graphs
- …