Bounds relating the weakly connected domination number to the total domination number and the matching number

AbstractLet G=(V,E) be a connected graph. A dominating set S of G is a weakly connected dominating set of G if the subgraph (V,E∩(S×V)) of G with vertex set V that consists of all edges of G incident with at least one vertex of S is connected. The minimum cardinality of a weakly connected dominating set of G is the weakly connected domination number, denoted γwc(G). A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt(G) of G. In this paper, we show that 12(γt(G)+1)≤γwc(G)≤32γt(G)−1. Properties of connected graphs that achieve equality in these bounds are presented. We characterize bipartite graphs as well as the family of graphs of large girth that achieve equality in the lower bound, and we characterize the trees achieving equality in the upper bound. The number of edges in a maximum matching of G is called the matching number of G, denoted α′(G). We also establish that γwc(G)≤α′(G), and show that γwc(T)=α′(T) for every tree T

Online Learning with Feedback Graphs: Beyond Bandits

We study a general class of online learning problems where the feedback is specified by a graph. This class includes online prediction with expert advice and the multi-armed bandit problem, but also several learning problems where the online player does not necessarily observe his own loss. We analyze how the structure of the feedback graph controls the inherent difficulty of the induced $T$-round learning problem. Specifically, we show that any feedback graph belongs to one of three classes: strongly observable graphs, weakly observable graphs, and unobservable graphs. We prove that the first class induces learning problems with $\widetilde\Theta(\alpha^{1/2} T^{1/2})$ minimax regret, where $\alpha$ is the independence number of the underlying graph; the second class induces problems with $\widetilde\Theta(\delta^{1/3}T^{2/3})$ minimax regret, where $\delta$ is the domination number of a certain portion of the graph; and the third class induces problems with linear minimax regret. Our results subsume much of the previous work on learning with feedback graphs and reveal new connections to partial monitoring games. We also show how the regret is affected if the graphs are allowed to vary with time

On the approximability of the maximum induced matching problem

In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3

Domination problems in directed graphs and inducibility of nets

In this thesis we discuss two topics: domination parameters and inducibility. In the first chapter, we introduce basic concepts, definitions, and a brief history for both types of problems. We will first inspect domination parameters in graphs, particularly independent domination in regular graphs and we answer a question of Goddard and Henning. Additionally, we provide some constructions for graphs regular graphs of small degree to provide lower bounds on the independent domination ratio of these classes of graphs. In Chapter 3 we expand our exploration of independent domination into the realm of directed graphs. We will prove several results including providing a fastest known algorithm for determining existence of an independent dominating set in directed graphs with minimum in degree at least one and period not eqeual to one. We also construct a set of counterexamples to the analogue of Vizing\u27s Conjecture for this setting. In the fourth chapter, we pivot from independent domination to split domination in directed graphs, where we introduce the split domination sequence. We will determine that almost all possible split domination sequences are realizable by some graphs, and state several open questions that would be of interest to continue on this field. In the fifth chapter we will provide a brief introduction to Flag Algebras, then determine the unique maximizer of induced net graphs in graphs of certain orders

Combinatorics of Castelnuovo-Mumford Regularity of Binomial Edge Ideals

Since the introduction of binomial edge ideals $J_{G}$ by Herzog et al. and independently Ohtani, there has been significant interest in relating algebraic invariants of the binomial edge ideal with combinatorial invariants of the underlying graph $G$. Here, we take up a question considered by Herzog and Rinaldo regarding Castelnuovo--Mumford regularity of block graphs. To this end, we introduce a new invariant $\nu(G)$ associated to any simple graph $G$, defined as the size of a particularly nice induced maximal matching of $\text{in}_{\text{lex}}(J_{G})$. We investigate the question of expressing $\nu(G)$ in terms of the combinatorics of $G$. We prove that $\nu(G)$ is the maximal total length of a certain collection of induced paths within $G$. We prove that for any graph $G$, $\nu(G) \leq \text{reg}(J_{G})-1$, and that the length of a longest induced path of $G$ is less than or equal to $\nu(G)$; this refines an inequality of Matsuda and Murai. We then investigate the question: when is $\nu(G) = \text{reg}(J_{G})-1$? We prove that equality holds when $G$ is closed; this gives a new characterization of a result of Ene and Zarojanu, and when $G$ is bipartite and $J_{G}$ is Cohen-Macaulay; this gives a new characterization of a result of Jayanathan and Kumar. For a block graph $G$ we prove that $\nu(G)$ admits a combinatorial characterization independent of any auxiliary choices, and we prove that $\nu(G) = \text{reg}(J_{G})-1$. This gives $\text{reg}(J_{G})$ a combinatorial interpretation for block graphs, and thus answers the question of Herzog and Rinaldo