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Michael A. Henning

Publications and source records attributed to Michael A. Henning.

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arXiv2015

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 $γ_{\min}(d)=\min\{ γ(G):G\in {\cal G}(d)\}$, $α_{\max}(d)=\max\{ α(G):G\in {\cal G}(d)\}$, $γ_{\min}^{\cal F}(d)=\min\{ γ(F):F\in {\cal F}(d)\}$, and $α_{\max}^{\cal F}(d)=\max\{ α(F):F\in {\cal F}(d)\}$ where $γ(G)$ is the domination number and $α(G)$ is the independence number of a graph $G$. Adapting results of Havel and Hakimi, Rao showed in 1979 that $α_{\max}(d)$ can be determined in polynomial time. We establish the existence of realizations $G\in {\cal G}(d)$ with $γ_{\min}(d)=γ(G)$, and $F_γ,F_α\in {\cal F}(d)$ with $γ_{\min}^{\cal F}(d)=γ(F_γ)$ and $α_{\max}^{\cal F}(d)=α(F_α)$ that have strong structural properties. This leads to an efficient algorithm to determine $γ_{\min}(d)$ for every given degree sequence $d$ with bounded entries as well as closed formulas for $γ_{\min}^{\cal F}(d)$ and $α_{\max}^{\cal F}(d)$.

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arXiv2015

Transversals in $4$-Uniform Hypergraphs

Let $H$ be a $3$-regular $4$-uniform hypergraph on $n$ vertices. The transversal number $τ(H)$ of $H$ is the minimum number of vertices that intersect every edge. Lai and Chang [J. Combin. Theory Ser. B 50 (1990), 129--133] proved that $τ(H) \le 7n/18$. Thomassé and Yeo [Combinatorica 27 (2007), 473--487] improved this bound and showed that $τ(H) \le 8n/21$. We provide a further improvement and prove that $τ(H) \le 3n/8$, which is best possible due to a hypergraph of order eight. More generally, we show that if $H$ is a $4$-uniform hypergraph on $n$ vertices and $m$ edges with maximum degree $Δ(H) \le 3$, then $τ(H) \le n/4 + m/6$, which proves a known conjecture. We show that an easy corollary of our main result is that the total domination number of a graph on $n$ vertices with minimum degree at least~4 is at most $3n/7$, which was the main result of the Thomassé-Yeo paper [Combinatorica 27 (2007), 473--487].

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arXiv2015

Matchings and Path Covers with applications to Domination in Graphs

Let $G$ be a graph with no isolated vertex. A matching in $G$ is a set of edges that are pairwise not adjacent in $G$, while the matching number, $α'(G)$, of $G$ is the maximum size of a matching in $G$. The path covering number, $\rm{pc}(G)$, of $G$ is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover. We show that if $G$ has order $n$, then $α'(G) + \frac{1}{2}\rm{pc}(G) \ge \frac{n}{2}$ and we provide a constructive characterization of the graphs achieving equality in this bound. It is known that $γ(G) \le α'(G)$ and $γ_t(G) \le α'(G) + \rm{pc}(G)$, where $γ(G)$ and $γ_t(G)$ denote the domination and the total domination number of $G$. As an application of our result on the matching and path cover numbers, we show that if $G$ is a graph with $δ(G) \ge 3$, then $γ_t(G) \le α'(G) + \frac{1}{2}(\rm{pc}(G) - 1)$, and this bound is tight. A set $S$ of vertices in $G$ is a neighborhood total dominating set of $G$ if it is a dominating set of $G$ with the property that the subgraph induced by the open neighborhood of the set $S$ has no isolated vertex. The neighborhood total domination number, $γ_{\rm nt}(G)$, is the minimum cardinality of a neighborhood total dominating set of $G$. We observe that $γ(G) \le γ_{\rm nt}(G) \le γ_t(G)$. As a further application of our result on the matching and path cover numbers, we show that if $G$ is a connected graph on at least six vertices, then $γ_{\rm nt}(G) \le α'(G) + \frac{1}{2}\rm{pc}(G)$ and this bound is tight.

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arXiv2014

Graphs with Large Disjunctive Total Domination Number

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $γ_t(G)$. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance $2$ from it. The disjunctive total domination number, $γ^d_t(G)$, is the minimum cardinality of such a set. We observe that $γ^d_t(G) \le γ_t(G)$. Let $G$ be a connected graph on $n$ vertices with minimum degree $δ$. It is known [J. Graph Theory 35 (2000), 21--45] that if $δ\ge 2$ and $n \ge 11$, then $γ_t(G) \le 4n/7$. Further [J. Graph Theory 46 (2004), 207--210] if $δ\ge 3$, then $γ_t(G) \le n/2$. We prove that if $δ\ge 2$ and $n \ge 8$, then $γ^d_t(G) \le n/2$ and we characterize the extremal graphs.

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arXiv2014

Disjunctive Total Domination in Graphs

Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $γ_t(G)$. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance2 from it. The disjunctive total domination number, $γ^d_t(G)$, is the minimum cardinality of such a set. We observe that $γ^d_t(G) \le γ_t(G)$. We prove that if $G$ is a connected graph of order$n \ge 8$, then $γ^d_t(G) \le 2(n-1)/3$ and we characterize the extremal graphs. It is known that if $G$ is a connected claw-free graph of order$n$, then $γ_t(G) \le 2n/3$ and this upper bound is tight for arbitrarily large$n$. We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if $G$ is a connected claw-free graph of order$n > 10$, then $γ^d_t(G) \le 4n/7$ and we characterize the graphs achieving equality in this bound.

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arXiv2014

Trees with Large Neighborhood Total Domination Number

In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [Opuscula Math. 31 (2011), 519--531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph $G$ is a dominating set $S$ in $G$ with the property that the subgraph induced by the open neighborhood of the set $S$ has no isolated vertex. The neighborhood total domination number, denoted by $\gnt(G)$, is the minimum cardinality of a NTD-set of $G$. Every total dominating set is a NTD-set, implying that $γ(G) \le \gnt(G) \le \gt(G)$, where $γ(G)$ and $\gt(G)$ denote the domination and total domination numbers of $G$, respectively. Arumugam and Sivagnanam posed the problem of characterizing the connected graphs $G$ of order $n \ge 3$ achieving the largest possible neighborhood total domination number, namely $\gnt(G) = \lceil n/2 \rceil$. A partial solution to this problem was presented by Henning and Rad [Discrete Applied Mathematics 161 (2013), 2460--2466] who showed that $5$-cycles and subdivided stars are the only such graphs achieving equality in the bound when $n$ is odd. In this paper, we characterize the extremal trees achieving equality in the bound when $n$ is even. As a consequence of this tree characterization, a characterization of the connected graphs achieving equality in the bound when $n$ is even can be obtained noting that every spanning tree of such a graph belongs to our family of extremal trees.

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arXiv2014

Induced 2-Regular Subgraphs in k-Chordal Cubic Graphs

We show that a cubic graph $G$ of order $n$ has an induced $2$-regular subgraph of order at least a) $\frac{n-2}{4-\frac{4}{k}}$, if $G$ has no induced cycle of length more than $k$, b) $\frac{5n+6}{8}$, if $G$ has no induced cycle of length more than $4$, and $n>6$, and c) $\left(\frac{1}{4}+ε\right)n$, if the independence number of $G$ is at most $\left(\frac{3}{8}-ε\right)n$. To show the second result we give a precise structural description of cubic $4$-chordal graphs.

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arXiv2014

Induced Cycles in Graphs

The maximum cardinality of an induced $2$-regular subgraph of a graph $G$ is denoted by $c_{\rm ind}(G)$. We prove that if $G$ is an $r$-regular graph of order $n$, then $c_{\rm ind}(G) \geq \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$ and we prove that if $G$ is a cubic claw-free graph on order $n$, then $c_{\rm ind}(G) > 13n/20$ and this bound is asymptotically best possible.

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arXiv2014

A characterization of hypergraphs that achieve equality in the Chvátal-McDiarmid Theorem

For $k \ge 2$, let $H$ be a $k$-uniform hypergraph on $n$ vertices and $m$ edges. The transversal number $τ(H)$ of $H$ is the minimum number of vertices that intersect every edge. Chvátal and McDiarmid [Combinatorica 12 (1992), 19--26] proved that $τ(H)\le ( n + \left\lfloor \frac k2 \right\rfloor m )/ ( \left\lfloor \frac{3k}2 \right\rfloor )$. When $k = 3$, the connected hypergraphs that achieve equality in the Chvátal-McDiarmid Theorem were characterized by Henning and Yeo [J. Graph Theory 59 (2008), 326--348]. In this paper, we characterize the connected hypergraphs that achieve equality in the Chvátal-McDiarmid Theorem for $k = 2$ and for all $k \ge 4$.

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arXiv2013

Total Transversals and Total Domination in Uniform Hypergraphs

The first three authors [European J. Combin. 33 (2012), 62--71] established a relationship between the transversal number and the domination number of uniform hypergraphs. In this paper, we establish a relationship between the total transversal number and the total domination number of uniform hypergraphs. We prove tight asymptotic upper bounds on the total transversal number in terms of the number of vertices, the number of edges, and the edge size.

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arXiv2013

Simultaneous Domination in Graphs

Let $F_1, F_2, ..., F_k$ be graphs with the same vertex set $V$. A subset $S \subseteq V$ is a simultaneous dominating set if for every $i$, $1 \le i \le k$, every vertex of $F_i$ not in $S$ is adjacent to a vertex in $S$ in $F_i$; that is, the set $S$ is simultaneously a dominating set in each graph $F_i$. The cardinality of a smallest such set is the simultaneous domination number. We present general upper bounds on the simultaneous domination number. We investigate bounds in special cases, including the cases when the factors, $F_i$, are $r$-regular or the disjoint union of copies of $K_r$. Further we study the case when each factor is a cycle.

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arXiv2011

Fair Domination in Graphs

A fair dominating set in a graph $G$ (or FD-set) is a dominating set $S$ such that all vertices not in $S$ are dominated by the same number of vertices from $S$; that is, every two vertices not in $S$ have the same number of neighbors in $S$. The fair domination number, $fd(G)$, of $G$ is the minimum cardinality of a FD-set. We present various results on the fair domination number of a graph. In particular, we show that if $G$ is a connected graph of order $n \ge 3$ with no isolated vertex, then $fd(G) \le n - 2$, and we construct an infinite family of connected graphs achieving equality in this bound. We show that if $G$ is a maximal outerplanar graph, then $fd(G) < 17n/19$. If $T$ is a tree of order $n \ge 2$, then we prove that $fd(T) \le n/2$ with equality if and only if $T$ is the corona of a tree.

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arXiv2011

Equality in a Linear Vizing-Like Relation that Relates the Size and Total Domination Number of a Graph

Let $G$ be a graph each component of which has order at least 3, and let $G$ have order $n$, size $m$, total domination number $γ_t$ and maximum degree $Δ(G)$. Let $Δ= 3$ if $Δ(G) = 2$ and $Δ= Δ(G)$ if $Δ(G) \ge 3$. It is known [J. Graph Theory 49 (2005), 285--290; J. Graph Theory 54 (2007), 350--353] that $m \le Δ(n- γ_t)$. In this paper we characterize the extremal graphs $G$ satisfying $m = Δ(n- γ_t)$.

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arXiv2010

Directed Domination in Oriented Graphs

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $γ(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $Γ_d(G)$, which is the maximum directed domination number $γ(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erdös [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. We extend this notion to directed domination of all graphs. If $α$ denotes the independence number of a graph $G$, we show that if $G$ is a bipartite graph, we show that $Γ_d(G) = α$. We present several lower and upper bounds on the directed domination number.

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arXiv2010

A Greedy Partition Lemma for Directed Domination

A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $γ(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $Γ_d(G)$, which is the maximum directed domination number $γ(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erdös [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this paper we prove a Greedy Partition Lemma for directed domination in oriented graphs. Applying this lemma, we obtain bounds on the directed domination number. In particular, if $α$ denotes the independence number of a graph $G$, we show that $α\le Γ_d(G) \le α(1+2\ln(n/α))$.

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