Smallest Domination Number and Largest Independence Number of Graphs and Forests with given Degree Sequence

For a sequence $d$ of non-negative integers, let ${\cal G}(d)$ and ${\cal F}(d)$ be the sets of all graphs and forests with degree sequence $d$, respectively. Let $\gamma_{\min}(d)=\min\{ \gamma(G):G\in {\cal G}(d)\}$, $\alpha_{\max}(d)=\max\{ \alpha(G):G\in {\cal G}(d)\}$, $\gamma_{\min}^{\cal F}(d)=\min\{ \gamma(F):F\in {\cal F}(d)\}$, and $\alpha_{\max}^{\cal F}(d)=\max\{ \alpha(F):F\in {\cal F}(d)\}$ where $\gamma(G)$ is the domination number and $\alpha(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $\alpha_{\max}(d)$ can be determined in polynomial time. We establish the existence of realizations $G\in {\cal G}(d)$ with $\gamma_{\min}(d)=\gamma(G)$, and $F_{\gamma},F_{\alpha}\in {\cal F}(d)$ with $\gamma_{\min}^{\cal F}(d)=\gamma(F_{\gamma})$ and $\alpha_{\max}^{\cal F}(d)=\alpha(F_{\alpha})$ that have strong structural properties. This leads to an efficient algorithm to determine $\gamma_{\min}(d)$ for every given degree sequence $d$ with bounded entries as well as closed formulas for $\gamma_{\min}^{\cal F}(d)$ and $\alpha_{\max}^{\cal F}(d)$

On an Annihilation Number Conjecture

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in the graph $G=\left(V,E\right)$. If $\alpha(G)+\mu(G)=\left\vert V\right\vert$, then $G$ is a K\"onig-Egerv\'ary graph. If $d_{1}\leq d_{2}\leq\cdots\leq d_{n}$ is the degree sequence of $G$, then the annihilation number $h\left(G\right)$ of $G$ is the largest integer $k$ such that $\sum\limits_{i=1}^{k}d_{i}\leq\left\vert E\right\vert$ (Pepper 2004, Pepper 2009). A set $A\subseteq V$ satisfying $\sum \limits_{a\in A} deg(a)\leq\left\vert E\right\vert$ is an annihilation set, if, in addition, $deg\left(v\right) +\sum\limits_{a\in A} deg(a)>\left\vert E\right\vert$, for every vertex $v\in V(G)-A$, then $A$ is a maximal annihilation set in $G$. In (Larson & Pepper 2011) it was conjectured that the following assertions are equivalent: (i) $\alpha\left(G\right) =h\left(G\right)$; (ii) $G$ is a K\"onig-Egerv\'ary graph and every maximum independent set is a maximal annihilating set. In this paper, we prove that the implication "(i) $\Longrightarrow$ (ii)" is correct, while for the opposite direction we provide a series of generic counterexamples. Keywords: maximum independent set, matching, tree, bipartite graph, K\"onig-Egerv\'ary graph, annihilation set, annihilation number.Comment: 17 pages, 11 figure

2-switch: transition and stability on graphs and forests

Given any two forests with the same degree sequence, we show in an algorithmic way that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests. We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequence. Then, we apply these results to conclude that the studied parameters have the interval property on those families.Comment: 23 pages, 2 figure

Irregular independence and irregular domination

If $A$ is an independent set of a graph $G$ such that the vertices in $A$ have different degrees, then we call $A$ an irregular independent set of $G$. If $D$ is a dominating set of $G$ such that the vertices that are not in $D$ have different numbers of neighbours in $D$, then we call $D$ an irregular dominating set of $G$. The size of a largest irregular independent set of $G$ and the size of a smallest irregular dominating set of $G$ are denoted by $\alpha_{ir}(G)$ and $\gamma_{ir}(G)$, respectively. We initiate the investigation of these two graph parameters. For each of them, we obtain sharp bounds in terms of basic graph parameters such as the order, the size, the minimum degree and the maximum degree, and we obtain Nordhaus-Gaddum-type bounds. We also establish sharp bounds relating the two parameters. Furthermore, we characterize the graphs $G$ with $\alpha_{ir}(G)=1$, we determine those that are planar, and we determine those that are outerplanar.Comment: 15 page

2-switch transition on unicyclic graphs and pseudoforest

In the present work we prove that given any two unicycle graphs (pseudoforests) that share the same degree sequence there is a finite sequence of 2-switches transforming one into the other such that all the graphs in the sequence are also unicyclic graphs (pseudoforests).Comment: 12 pages. arXiv admin note: text overlap with arXiv:2004.1116

Weighted Upper Edge Cover: Complexity and Approximability

Optimization problems consist of either maximizing or minimizing an objective function. Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution). Such "flipping" of the objective function was done for many classical optimization problems. For example, Minimum Vertex Cover becomes Maximum Minimal Vertex Cover, Maximum Independent Set becomes Minimum Maximal Independent Set and so on. In this paper, we propose to study the weighted version of Maximum Minimal Edge Cover called Upper Edge Cover, a problem having application in the genomic sequence alignment. It is well-known that Minimum Edge Cover is polynomial-time solvable and the "flipped" version is NP-hard, but constant approximable. We show that the weighted Upper Edge Cover is much more difficult than Upper Edge Cover because it is not $O(\frac{1}{n^{1/2-\varepsilon}})$ approximable, nor $O(\frac{1}{\Delta^{1-\varepsilon}})$ in edge-weighted graphs of size $n$ and maximum degree $\Delta$ respectively. Indeed, we give some hardness of approximation results for some special restricted graph classes such as bipartite graphs, split graphs and $k$-trees. We counter-balance these negative results by giving some positive approximation results in specific graph classes.Comment: 19 pages, 4 figure

The matching number of tree and bipartite degree sequences

We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible values. For bipartite degree sequences, we show the existence of realizations with a restricted structure, which allows to derive an analogue of the Gale-Ryser Theorem characterizing bipartite degree sequences. More precisely, we show that a bipartite degree sequence has a realization with a certain matching number if and only if a cubic number of inequalities similar to those in the Gale-Ryser Theorem are satisfied. For tree degree sequences as well as for bipartite degree sequences, the possible values of the matching number form intervals

Equating two maximum degrees

Given a graph $G$, we would like to find (if it exists) the largest induced subgraph $H$ in which there are at least $k$ vertices realizing the maximum degree of $H$. This problem was first posed by Caro and Yuster. They proved, for example, that for every graph $G$ on $n$ vertices we can guarantee, for $k = 2$, such an induced subgraph $H$ by deleting at most $2\sqrt{n}$ vertices, but the question if $2\sqrt{n}$ is best possible remains open. Among the results obtained in this paper we prove that: 1. For every graph $G$ on $n \geq 4$ vertices we can delete at most $\lceil \frac{- 3 + \sqrt{ 8n- 15}}{2 } \rceil$ vertices to get an induced subgraph $H$ with at least two vertices realizing $\Delta(H)$, and this bound is sharp, solving the problems left open by Caro and Yuster. 2.For every graph $G$ with maximum degree $\Delta \geq 1$ we can delete at most $\lceil \frac{ -3 + \sqrt{8\Delta +1}}{2 } \rceil$ vertices to get an induced subgraph $H$ with at least two vertices realizing $\Delta(H)$, and this bound is sharp. 3. Every graph $G$ with $\Delta(G) \leq 2$ and least $2k - 1$ vertices (respectively $2k - 2$ vertices if k is even) contains an induced subgraph $H$ in which at least $k$ vertices realise $\Delta(H)$, and these bound are sharp

Lower bounds for independence and kk-independence number of graphs using the concept of degenerate degrees

Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\zeta(v)$ as $\zeta(v)={\max}_{H: v\in H}~\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show that the degenerate degree sequence of any graph can be determined by an efficient algorithm. A $k$-independent set in $G$ is any set $S$ of vertices such that $\Delta(G[S])\leq k$. The largest cardinality of any $k$-independent set is denoted by $\alpha_k(G)$. For $k\in \{1, 2, 3\}$, we prove that $\alpha_{k-1}(G)\geq {\sum}_{v\in G} \min \{1, 1/(\zeta(v)+(1/k))\}$. Using the concept of cheap vertices we strengthen our bound for the independence number. The resulting lower bounds improve greatly the famous Caro-Wei bound and also the best known bounds for $\alpha_1(G)$ and $\alpha_2(G)$ for some families of graphs. We show that the equality in our bound for independence number happens for a large class of graphs. Our bounds are achieved by Cheap-Greedy algorithms for $\alpha_k(G)$ which are designed by the concept of cheap sets. At the end, a bound for $\alpha_k(G)$ is presented, where $G$ is a forest and $k$ an arbitrary non-negative integer

A logician's view of graph polynomials

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we distinguish between the graph theoretic (semantic) and the algebraic (syntactic) meaning of graph polynomials. We discuss how to represent and compare graph polynomials by their distinctive power. We introduce the class of graph polynomials definable using Second Order Logic which comprises virtually all examples of graph polynomials with a fixed finite set of indeterminates. Finally we show that the location of zeros and stability of graph polynomials is not a semantic property. The paper emphasizes a model theoretic view and gives a unified exposition of classical results in algebraic combinatorics together with new and some of our previously obtained results scattered in the graph theoretic literature.Comment: 46 pages, 2 figures, Expanded version of invited lecture at WOLLIC 2016 (Workshop on Logic, Language, Information and Computation, Puebla, Mexico, 2016), Revised version May 5, 201