arXiv subjects

Michael A. Henning

Publications and source records attributed to Michael A. Henning.

At least 19 recordsLinked to original sources

arXiv2026

On the Relationships between Domination, Isolation, and Packing

We consider the relationships between the domination number of graph, denoted $γ$, and the distance-$2$ domination number, denoted $γ_2$, and three parameters that lie between them: the packing number, denoted $ρ$, the lower packing number, denoted $ρ_L$, and the isolation number, denoted $ι$. There has been recent attention on the question of whether $γ/ρ$ is bounded or unbounded for various families of graphs. We consider similar questions for the ratios of the five parameters. In particular we show that, while $γ/ρ_L$ is unbounded in trees, it holds that $ι/γ_2$ is less than $2$ for all trees. Further, $γ/ρ_L$ is at most $3$ in interval graphs, at most~$4$ in permutation graphs, and at most $5$ in general asteroidal-triple-free graphs. We also show that every tree has a set of vertices that is both isolating and a packing, and characterize trees where $ρ=ρ_L$.

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arXiv2026

Bounds on the game isolation number and exact values for paths and cycles

The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $ι_{\rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $ι_{\rm g}'(G)$. In this paper, $ι_{\rm g}(C_n)$, $ι_{\rm g}(P_n)$, $ι_{\rm g}'(C_n)$, and $ι_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $ι_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $ι_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $ι_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $ι_{\rm g}(G) = ι_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds.

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arXiv2026

Independence, induced subgraphs, and domination in $K_{1,r}$-free graphs

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $α_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with \emph{chromatic number} at most~$k$, we prove that if $G$ is $K_{1,r}$-free, then $α_{\mathcal{F}}(G) \le (r-1)kγ(G)$, where $γ(G)$ is the \emph{domination number}. When $\mathcal{F}$ is the family of empty graphs, this bound simplifies to $α(G) \le 2γ(G)$ for $K_{1,3}$-free (claw-free) graphs, where $α(G)$ is the \emph{independence number} of $G$. For $d$-regular graphs, this is further refined to the bound $α(G) \le 2\left(\frac{d+1}{d+2}\right)γ(G)$, which is tight for $d \in \{2, 3, 4\}$. Using Ramsey theory, we extend this framework to edge-hereditary graph families, showing that for $K_{1,r}$-free graphs, we have $α_{\mathcal{F}}(G) \le r(K_r, \mathcal{F^*})γ(G)$, where $\mathcal{F^*}$ is the set of graphs not in $\mathcal{F}$. Specializing to $K_q$-free graphs, we show $α_{\mathcal{F}}(G) \le (r(K_q, K_r) - 1)γ(G)$. Finally, for the \emph{$k$-independence number} $α_k(G)$, we prove that if $G$ is $K_{1,r}$-free with order $n$ and minimum degree $δ\ge k+1$, \[ α_k(G) \le \left( \frac{(r-1)(k+1)}{δ- k + (r-1)(k+1)} \right) n, \] and this bound is sharp for all parameters.

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arXiv2026

Total isolation game in graphs

The total isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $S$ is the set of already played vertices, then a vertex can be selected only if it is adjacent to a vertex that belongs to a (nontrivial) component of the graph $G - N_G(S)$ of order at least $2$ or a vertex that is isolated in $G - N_G(S)$ and belongs to the set $S$, where $N_G(S)$ is the set of vertices adjacent to a vertex in $S$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game total isolation number $ι_{\rm gt}(G)$ is the number of moves in the Dominator-start game where both players play optimally. We prove that if $G$ is a connected graph of order $n \ge 3$, then $ι_{\rm gt}(G) < \frac{5}{6}n$. Furthermore if $G$ has minimum degree at least $2$, then we prove that $ι_{\rm gt}(G) \le \frac{3}{4}n$. More generally, if $G$ is a connected graph of order $n \ge 3$ with minimum degree $δ$ where $δ\ge 2$, then we prove that $ι_{\rm gt}(G) \le \left( \frac{2δ-1}{3δ-2} \right) n$. Among other results it is proved that if $G$ is a graph of order $n$ with diameter $2$, then $ι_{\rm gt}(G) \le \frac{2}{3}n$.

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arXiv2025

A proof of the $\frac{3}{8}$-conjecture for independent domination in cubic graphs

A set $S$ of vertices in a graph $G$ is a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in~$S$. An independent dominating set in $G$ is a dominating set of $G$ with the additional property that it is an independent set. The domination number, $γ(G)$, and the independent domination number, $i(G)$, are the minimum cardinalities among all dominating sets and independent dominating sets in $G$, respectively. By definition, $γ(G) \le i(G)$ for all graphs $G$. Let $G$ be a connected cubic graph of order~$n$. In 1996 Reed [Combin.\ Probab.\ Comput.\ 5 (1996), 277--295] proved a breakthrough result that $γ(G) \le \frac{3}{8}n$. We prove the stronger result that if $G$ is different from $K_{3,3}$ and the $5$-prism $C_5 \, \Box \, K_2$, then $i(G) \le \frac{3}{8}n$. This proves a known conjecture. The bound is tight in the sense that there are infinite families of connected cubic graphs that achieve equality in this bound.

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arXiv2025

Identifying codes in graphs of given maximum degree: Characterizing trees

An identifying code of a closed-twin-free graph $G$ is a dominating set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhoods and $S$. It was conjectured that there exists an absolute constant $c$ such that for every connected graph $G$ of order $n$ and maximum degree $Δ$, the graph $G$ admits an identifying code of size at most $( \frac{Δ-1}Δ )n +c$. We provide significant support for this conjecture by exactly characterizing every tree requiring a positive constant $c$ together with the exact value of the constant. Hence, proving the conjecture for trees. For $Δ=2$ (the graph is a path or a cycle), it is long known that $c=3/2$ suffices. For trees, for each $Δ\ge 3$, we show that $c=1/Δ\le 1/3$ suffices and that $c$ is required to have a positive value only for a finite number of trees. In particular, for $Δ= 3$, there are 12 trees with a positive constant $c$ and, for each $Δ\ge 4$, the only tree with positive constant $c$ is the $Δ$-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtilä. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022]. We remark that there are infinitely many trees for which the bound is tight when $Δ=3$; for every $Δ\ge 4$, we construct an infinite family of trees of order $n$ with identification number very close to the bound, namely $\left( \frac{Δ-1+\frac{1}{Δ-2}}{Δ+\frac{2}{Δ-2}} \right) n > (\frac{Δ-1}Δ ) n -\frac{n}{Δ^2}$. Furthermore, we also give a new tight upper bound for identification number on trees by showing that the sum of the domination and identification numbers of any tree $T$ is at most its number of vertices.

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arXiv2025

Spreading in claw-free cubic graphs

Let $p \in \mathbb{N}$ and $q \in \mathbb{N} \cup \lbrace \infty \rbrace$. We study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of blue vertices, with all remaining vertices colored white. If a white vertex~$v$ has at least~$p$ blue neighbors and at least one of these blue neighbors of~$v$ has at most~$q$ white neighbors, then by the spreading color change rule the vertex~$v$ is recolored blue. The initial set $S$ of blue vertices is a $(p,q)$-spreading set for $G$ if by repeatedly applying the spreading color change rule all the vertices of $G$ are eventually colored blue. The $(p,q)$-spreading set is a generalization of the well-studied concepts of $k$-forcing and $r$-percolating sets in graphs. For $q \ge 2$, a $(1,q)$-spreading set is exactly a $q$-forcing set, and the $(1,1)$-spreading set is a $1$-forcing set (also called a zero forcing set), while for $q = \infty$, a $(p,\infty)$-spreading set is exactly a $p$-percolating set. The $(p,q)$-spreading number, $σ_{(p,q)}(G)$, of $G$ is the minimum cardinality of a $(p,q)$-spreading set. In this paper, we study $(p,q)$-spreading in claw-free cubic graphs. While the zero-forcing number of claw-free cubic graphs was studied earlier, for each pair of values $p$ and $q$ that are not both $1$ we either determine the $(p,q)$-spreading number of a claw-free cubic graph $G$ or show that $σ_{(p,q)}(G)$ attains one of two possible values.

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arXiv2025

On the isolation number of graphs with minimum degree four

An isolating set in a graph $G$ is a set $S$ of vertices such that removing $S$ and its neighborhood leaves no edge. The isolation number $ι(G)$ of $G$ (also known as the vertex-edge domination number) is the minimum size among all isolating sets of $G$. We provide a technique for proving upper bounds on this parameter for graphs with a given minimum degree. For example, we show that if $G$ has order~$n$ and minimum degree at least~$4$, then $ι(G) \le 13n/41$, and if $G$ is also triangle-free, then $ι(G) \le 3n/10$.

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arXiv2025

On $k$-coalition in graphs: bounds and exact values

Given a graph $G=\big{(}V(G),E(G)\big{)}$, a set $S\subseteq V(G)$ is called a $k$-dominating set if every vertex in $V(G)\setminus S$ has at least $k$ neighbors in $S$. Two disjoint sets $A,B\subset V(G)$ form a $k$-coalition in $G$ if neither set is a $k$-dominating set in $G$ but their union $A\cup B$ is a $k$-dominating set. A partition $Ω$ of $V(G)$ is a $k$-coalition partition if each set in $Ω$ is either a $k$-dominating set of cardinality $k$ or forms a $k$-coalition with another set in $Ω$. The $k$-coalition number $C_{k}(G)$ equals the maximum cardinality of a $k$-coalition partition of $G$. In this work, we give general upper and lower bounds on this parameter. In particular, we show that if $G$ has minimum degree $δ\ge 2$ and maximum degree $Δ\ge 4 \lfloor δ/2 \rfloor$, then $C_{2}(G) \leq (Δ-2\lfloor δ/2 \rfloor+1)(\lfloor δ/2 \rfloor+1) + \lceil δ/2 \rceil+1$, and this bound is sharp. If $T$ is a tree of order~$n \ge 2$, then we prove the upper bound $C_{2}(T) \leq \big\lfloor \frac{n}{2}\big\rfloor+1$ and we characterize the extremal trees achieving equality in this bound. We determine the exact value of $C_{k}(G)$ for any cubic graph $G$ and $k\geq2$. Finally, we give the exact value of $C_{k}$ for any complete bipartite graph, which completes a partial result and resolves an issue from an earlier paper.

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arXiv2025

Secure domination in $P_5$-free graphs

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of $G$ with the additional property that for every vertex $v \in V(G) \setminus S$, there exists a neighbor $u$ of $v$ in $S$ such that $(S \setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. The secure domination number of $G$, denoted by $γ_s(G)$, is the minimum cardinality of a secure dominating set of $G$. We prove that if $G$ is a $P_5$-free graph, then $γ_s(G) \le \frac{3}{2}α(G)$, where $α(G)$ denotes the independence number of $G$. We further show that if $G$ is a connected $(P_5, H)$-free graph for some $H \in \{ P_3 \cup P_1, K_2 \cup 2K_1, ~\text{paw},~ C_4\}$, then $γ_s(G)\le \max\{3,α(G)\}$. We also show that if $G$ is a $(P_3 \cup P_2)$-free graph, then $γ_s(G)\le α(G)+1$.

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arXiv2025

On polluted bootstrap percolation in Cartesian grids

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we continue the study of polluted bootstrap percolation introduced and studied by Gravner and McDonald [Bootstrap percolation in a polluted environment. J.\ Stat\ Physics 87 (1997) 915--927] where in this variant some vertices are permanently in the non-infected state. We study an extremal (combinatorial) version of the bootstrap percolation problem in a polluted environment, where our main focus is on the class of grid graphs, that is, the Cartesian product $P_m \square P_n$ of two paths $P_m$ and $P_n$ on $m$ and $n$ vertices, respectively. Given a number of polluted vertices in a Cartesian grid we establish a closed formula for the minimum $2$-neighbor bootstrap percolation number of the polluted grid, and obtain a lower bound for the other extreme.

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arXiv2025

Paired domination in trees: A linear algorithm and asymptotic normality

A set $S$ of vertices in a graph $G$ is a paired dominating set if every vertex of $G$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, $γ_{\mathrm{pr}}(G)$, of $G$ is the minimum cardinality of a paired dominating set of $G$. We present a linear algorithm for computing the paired domination number of a tree. As an application of our algorithm, we prove that the paired domination number is asymptotically normal in a random rooted tree of order $n$ generated by a conditioned Galton-Watson process as $n\to\infty$. In particular, we have found that the paired domination number of a random Cayley tree of order $n$, where each tree is equally likely, is asymptotically normal with expectation approaching $(0.5177\ldots)n$.

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arXiv2025

Paired domination in graphs with minimum degree four

A set $S$ of vertices in a graph $G$ is a paired dominating set if every vertex of $G$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ admits a perfect matching. The minimum cardinality of a paired dominating set of $G$ is the paired domination number $\gpr(G)$ of $G$. We show that if $G$ is a graph of order~$n$ and $δ(G) \ge 4$, then $\gpr(G) \le \frac{10}{17}n < 0.5883 n$.

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arXiv2025

Disjunctive domination in maximal outerplanar graphs

A disjunctive dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G)\setminus D$ has a neighbor in $D$ or has at least two vertices in $D$ at distance $2$ from it. The disjunctive domination number of $G$, denoted by $γ_2^d(G)$, is the minimum cardinality of a disjunctive dominating set of $G$. In this paper, we show that if $G$ is a maximal outerplanar graph of order $n \ge 7$ with $k$ vertices of degree $2$, then $γ_2^d(G)\le \lfloor\frac{2}{9}(n+k)\rfloor$, and this bound is sharp.

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arXiv2024

An exploration of the balance game

The balance game is played on a graph $G$ by two players, Admirable (A) and Impish (I), who take turns selecting unlabeled vertices of $G$. Admirable labels the selected vertices by $0$ and Impish by $1$, and the resulting label on any edge is the sum modulo $2$ of the labels of the vertices incident to that edge. Let $e_0$ and $e_1$ denote the number of edges labeled by $0$ and $1$ after all the vertices are labeled. The discrepancy in the balance game is defined as $d = e_1 - e_0$. The two players have opposite goals: Admirable attempts to minimize the discrepancy $d$ while Impish attempts to maximize $d$. When (A) makes the first move in the game, the (A)-start game balance number, $b^A_g(G)$, is the value of $d$ when both players play optimally, and when (I) makes the first move in the game, the (I)-start game balance number, $b^I_g(G)$, is the value of $d$ when both players play optimally. Among other results, we show that if $G$ has order $n$, then $-\log_2(n) \le b^A_g(G) \le \frac{n}{2}$ if $n$ is even and $0 \le b^A_g(G) \le \frac{n}{2} + \log_2(n)$ if $n$ is odd. Moreover we show that $b^A_g(G) + b^I_g(\overline{G}) = \lfloor n/2 \rfloor$.

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arXiv2024

Progress towards the two-thirds conjecture on locating-total dominating sets

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $γ^L_t(G)$. It has been conjectured that $γ^L_t(G)\leq\frac{2n}{3}$ holds for every twin-free graph $G$ of order $n$ without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs and subcubic graphs.

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arXiv2024

Identifying codes in triangle-free graphs of bounded maximum degree

An $\textit{identifying code}$ of a closed-twin-free graph $G$ is a set $S$ of vertices of $G$ such that any two vertices in $G$ have a distinct intersection between their closed neighborhood and $S$. It was conjectured that there exists a constant $c$ such that for every connected closed-twin-free graph $G$ of order $n$ and maximum degree $Δ$, the graph $G$ admits an identifying code of size at most $\left( \frac{Δ-1}Δ \right) n+c$. In [D. Chakraborty, F. Foucaud, M. A. Henning, and T. Lehtilä. Identifying codes in graphs of given maximum degree: Characterizing trees. arXiv preprint arXiv:2403.13172, 2024], we proved the conjecture for all trees. In this article, we show that the conjecture holds for all triangle-free graphs, with the same list of exceptional graphs needing $c>0$ as for trees: for $Δ\ge 3$, $c=1/3$ suffices and there is only a set of 12 trees requiring $c>0$ for $Δ=3$, and when $Δ\ge 4$ this set is reduced to the $Δ$-star only. Our proof is by induction, whose starting point is the above result for trees. Along the way, we prove a generalized version of Bondy's theorem on induced subsets [J. A. Bondy. Induced subsets. Journal of Combinatorial Theory, Series B, 1972] that we use as a tool in our proofs. We also use our main result for triangle-free graphs, to prove the upper bound $\left( \frac{Δ-1}Δ \right) n+1/Δ+4t$ for graphs that can be made triangle-free by the removal of $t$ edges.

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arXiv2024

Identifying open codes in trees and 4-cycle-free graphs of given maximum degree

An identifying open code of a graph $G$ is a set $S$ of vertices that is both a separating open code (that is, $N_G(u) \cap S \ne N_G(v) \cap S$ for all distinct vertices $u$ and $v$ in $G$) and a total dominating set (that is, $N(v) \cap S \ne \emptyset$ for all vertices~$v$ in $G$). Such a set exists if and only if the graph $G$ is open twin-free and isolate-free; and the minimum cardinality of an identifying open code in an open twin-free and isolate-free graph $G$ is denoted by $γ^{\rm {\small IOC}}(G)$. We study the smallest size of an identifying open code of a graph, in relation with its order and its maximum degree. For $Δ$ a fixed integer at least $3$, if $G$ is a connected graph of order $n \ge 5$ that contains no $4$-cycle and is open twin-free with maximum degree bounded above by $Δ$, then we show that $γ^{\rm {\small IOC}}(G) \le \left( \frac{2Δ- 1}Δ \right) n$, unless $G$ is obtained from a star $K_{1,Δ}$ by subdividing every edge exactly once. Moreover, we show that the bound is best possible by constructing graphs that reach the bound.

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