Strongly well-covered graphs

AbstractA graph is well-covered if every maximal independent set is a maximum independent set. A strongly well-covered graph G has the additional property that G-e is also well-covered for every line e in G. Hence, the strongly well-covered graphs are a subclass of the well-covered graphs. We characterize strongly well-covered graphs with independence number two and determine a parity condition for strongly well-covered graphs with independence number three. More generally, we show that a strongly well-covered graph (with more than four points) is 3-connected and has minimum degree at least four

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

Graphs with the strong Havel-Hakimi property

The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel-Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel-Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.Comment: 7 pages, 3 figure

Independent sets and cuts in large-girth regular graphs

We present a local algorithm producing an independent set of expected size $0.44533n$ on large-girth 3-regular graphs and $0.40407n$ on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with $1.34105n$ edges on large-girth 3-regular graphs. These decrease the gaps between the best known upper and lower bounds from $0.0178$ to $0.01$, from $0.0242$ to $0.0123$ and from $0.0724$ to $0.0616$, respectively. We are using local algorithms, therefore, the method also provides upper bounds for the fractional coloring numbers of $1 / 0.44533 \approx 2.24554$ and $1 / 0.40407 \approx 2.4748$ and fractional edge coloring number $1.5 / 1.34105 \approx 1.1185$. Our algorithms are applications of the technique introduced by Hoppen and Wormald

Degree Sequence Index Strategy

We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by which to bound graph invariants by certain indices in the ordered degree sequence. As an illustration of the DSI strategy, we show how it can be used to give new upper and lower bounds on the $k$-independence and the $k$-domination numbers. These include, among other things, a double generalization of the annihilation number, a recently introduced upper bound on the independence number. Next, we use the DSI strategy in conjunction with planarity, to generalize some results of Caro and Roddity about independence number in planar graphs. Lastly, for claw-free and $K_{1,r}$-free graphs, we use DSI to generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester