A lower bound on independence in terms of degrees

We prove a new lower bound on the independence number of a simple connected graph in terms of its degrees

On F-independence in graphs

Let F be a set of graphs and for a graph G let F(G) and F (G) denote the maximum order of an induced subgraph of G which does not contain a graph in F as a subgraph and which does not contain a graph in F as an induced subgraph, respectively. Lower bounds on F (G) and F (G) and algorithms realizing them are presented

Independence in connected graphs

We prove that if $G=(V_G,E_G)$ is a finite, simple, and undirected graph with $\kappa$ components and independence number $\alpha(G)$, then there exist a positive integer $k\in \mathbb{N}$ and a function $f:V_G\to \mathbb{N}_0$ with non-negative integer values such that $f(u)\leq d_G(u)$ for $u\in V_G$, $\alpha(G)\geq k\geq \sum\limits_{u\in V_G}\frac{1}{d_G(u)+1-f(u)},$ and $\sum\limits_{u\in V_G}f(u)\geq 2(k-\kappa).$ This result is a best-possible improvement of a result due to Harant and Schiermeyer (On the independence number of a graph in terms of order and size, {\it Discrete Math.} {\bf 232} (2001), 131-138) and implies that $\frac{\alpha(G)}{n(G)}\geq \frac{2}{\left(d(G)+1+\frac{2}{n(G)}\right)+\sqrt{\left(d(G)+1+\frac{2}{n(G)}\right)2-8}}$ for connected graphs $G$ of order $n(G)$, average degree $d(G)$, and independence number $\alpha(G)$

Independence, odd girth, and average degree

We prove several best-possible lower bounds in terms of the order and the average degree for the independence number of graphs which are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle-free graphs of maximum degree at most $3$ due to Heckman and Thomas [A New Proof of the Independence Ratio of Triangle-Free Cubic Graphs, {\it Discrete Math.} {\bf 233} (2001), 233-237] to arbitrary triangle-free graphs. For connected triangle-free graphs of order $n$ and size $m$, our result implies the existence of an independent set of order at least $(4n-m-1)/7$

In the complement of a dominating set

A set D of vertices of a graph G=(V,E) is a dominating set, if every vertex of D\V has at least one neighbor that belongs to D. The disjoint domination number of a graph G is the minimum cardinality of two disjoint dominating sets of G. We prove upper bounds for the disjoint domination number for graphs of minimum degree at least 2, for graphs of large minimum degree and for cubic graphs.A set T of vertices of a graph G=(V,E) is a total dominating set, if every vertex of G has at least one neighbor that belongs to T. We characterize graphs of minimum degree 2 without induced 5-cycles and graphs of minimum degree at least 3 that have a dominating set, a total dominating set, and a non-empty vertex set that are disjoint.A set I of vertices of a graph G=(V,E) is an independent set, if all vertices in I are not adjacent in G. We give a constructive characterization of trees that have a maximum independent set and a minimum dominating set that are disjoint and we show that the corresponding decision problem is NP-hard for general graphs. Additionally, we prove several structural and hardness results concerning pairs of disjoint sets in graphs which are dominating, independent, or both. Furthermore, we prove lower bounds for the maximum cardinality of an independent set of graphs with specifed odd girth and small average degree.A connected graph G has spanning tree congestion at most s, if G has a spanning tree T such that for every edge e of T the edge cut defined in G by the vertex sets of the two components of T-e contains at most s edges. We prove that every connected graph of order n has spanning tree congestion at most n^(3/2) and we show that the corresponding decision problem is NP-hard