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Discrete Mathematics & Theoretical Computer Science |
vol. 28:2
1. Approximations for Fault-Tolerant Total and Partial Positive Influence Domination
In $\textit{total domination}$, given a graph $G=(V,E)$, we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ has at least one neighbor in $S$. We define a $\textit{fault-tolerant}$ version of total domination, where we require any node in $V \setminus S$ to have at least $m$ neighbors in $S$. Let $Δ$ denote the maximum degree in $G$. We prove a first $1 + \ln(Δ+ m - 1)$ approximation for fault-tolerant total domination. We also consider fault-tolerant variants of the weighted $\textit{partial positive influence dominating set}$ problem, where we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ is either a member of $S$ or the sum of weights of its incident edges leading to nodes in $S$ is at least half of the sum of weights over all its incident edges. We prove the first logarithmic approximations for the simple, total, and connected variants of this problem. To prove the result for the connected case, we extend the general approximation framework for non-submodular functions from integer-valued to fractional-valued functions, which we believe is of independent interest.
Section: Discrete Algorithms
2. Permutation-based Strategies for Labeled Chip-Firing on $k$-ary Trees
Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed $k$-ary tree where we place $k^n$ chips labeled $0,1,\dots, k^n-1$ on the root for some nonnegative integer $n$, and we say a vertex $v$ can fire if it has at least $k$ chips. When a vertex fires, we select $k$ labeled chips and send the $i$th smallest chip among them to its $i$th leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of $1, 2, \dots, n$. We then express the stable configuration as a permutation of $0,1, 2, \dots, k^n-1$ and explore its properties, such as the number of inversions and descents.
Section: Combinatorics
3. Planar-Toroidal Decomposition of $K_{12}$
In 1978, Anderson and White asked whether there is a decomposition of $K_{12}$ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if $G$ is planar of order 12 and $H\subseteq\overline{G}$ is toroidal, then $H$ has at least two fewer edges than $\overline{G}$. A computer search found all 123 unique pairs $\left(G,H\right)$ that make this an equality.
Section: Graph Theory
4. The AVD-total chromatic number of fullerene molecular graphs
An \textit{AVD-$k$-total coloring} of a simple graph $G$ is a mapping $\pi:V(G) \cup E(G) \to \{1,\ldots,k\}$, with $k \geq 1$ such that: for each pair of adjacent or incident elements $x,y \in V(G) \cup E(G)$, $\pi(x) \neq \pi(y)$; and for each pair of adjacent vertices $x,y \in V(G)$, sets $\{\pi(x)\} \cup \{\pi(xv) \mid xv \in E(G), v \in V(G)\}$ and $\{\pi(y)\} \cup \{\pi(yv)\mid yv \in E(G), v \in V(G)\}$ are distinct. The \textit{AVD-total chromatic number}, denoted by $\chi''_{a}(G)$ is the smallest $k$ for which $G$ admits an AVD-$k$-total-coloring. We consider a conjecture proposed in 2010 in the thesis of Jonathan Hulgan that any graph~$G$ with maximum vertex degree 3 has $\chi''_{a}(G) \leq 5$. As positive evidence, we prove that several molecular graphs known as fullerene graphs have AVD-total chromatic number equal to 5.
Section: Graph Theory
5. On induced subgraphs of $H(n,3)$ with maximum degree $1$
In this paper, we consider induced subgraphs of the Hamming graph $H(n,3)$. We show that if $U \subseteq \mathbb{Z}_3^n$ and $U$ induces a subgraph of $H(n,3)$ with maximum degree at most $1$ then 1. If $U$ is disjoint from a maximum size independent set of $H(n,3)$ then $|U| \leq 3^{n-1}+1$. Moreover, all such $U$ with size $3^{n-1}+1$ are isomorphic to each other. 2. For $n \geq 6$, there exists such a $U$ with size $|U| = 3^{n-1}+18$ and this is optimal for $n = 6$. 3. If $U \cap \{x, x+e_1, x+2e_1\} \ne ϕ$ for all $x \in \mathbb{Z}_3^n$ then $|U| \leq 3^{n-1} + 81$.
Section: Graph Theory
6. Vertex-transitive nut graph order-degree existence problem
A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy $4 \mid d$, $d \ge 4$, $2 \mid n$ and $n \ge d + 4$; or $d \equiv 2 \pmod 4$, $d \ge 6$, $4 \mid n$ and $n \ge d + 6$. Here, we prove that for each such $n$ and $d$, there exists a $d$-regular Cayley nut graph of order $n$. As a direct consequence, we obtain all the pairs $(n, d)$ for which there is a $d$-regular vertex-transitive (resp. Cayley) nut graph of order $n$.
Section: Graph Theory
7. Degree Realization by Bipartite Multigraphs
The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing sequences that can be realized by a bipartite graph has two variants. In the simpler one, termed BDR$^P$, the partition of the sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are also two variants. For BDR$^P$, where the partition is given as part of the input, a characterization was known for determining whether there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most $r$. We present a characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most $t$. We show that optimizing these two measures may lead to different realizations, and that optimizing by one measure may increase the other substantially. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. Moreover, we show that […]
Section: Graph Theory
8. Limit theorems for fixed point biased permutations avoiding a pattern of length three
We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution towards favoring permutations with more or fewer fixed points than is typical under the uniform distribution. One case we study features a phase transition where the limiting distribution changes abruptly from negative binomial to Rayleigh to normal depending on the bias parameter.
Section: Combinatorics
9. Sometimes Two Irrational Guards are Needed
In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.
Section: Automata, Logic and Semantics
10. When Many Trees Go to War: On Sets of Phylogenetic Trees With Almost No Common Structure
It is known that any two trees on the same $n$ leaves can be displayed by a network with $n-2$ reticulations, and there are two trees that cannot be displayed by a network with fewer reticulations. But how many reticulations are needed to display multiple trees? For any set of $t$ trees on $n$ leaves, there is a trivial network with $(t - 1)n$ reticulations that displays them. To do better, we have to exploit common structure of the trees to embed non-trivial subtrees of different trees into the same part of the network. In this paper, we show that for $t \in o(\sqrt{\lg n})$, there is a set of $t$ trees with virtually no common structure that could be exploited. More precisely, we show for any $t\in o(\sqrt{\lg n})$, there are $t$ trees such that any network displaying them has $(t-1)n - o(n)$ reticulations. For $t \in o(\lg n)$, we obtain a slightly weaker bound. We also prove that already for $t = c\lg n$, for any constant $c > 0$, there is a set of $t$ trees that cannot be displayed by a network with $o(n \lg n)$ reticulations, matching up to constant factors the known upper bound of $O(n \lg n)$ reticulations sufficient to display \emph{all} trees with $n$ leaves. These results are based on simple counting arguments and extend to unrooted networks and trees.
Section: Combinatorics
11. Probabilistic Counters for Privacy Preserving Data Aggregation
Probabilistic counters are well-known tools often used for space-efficient set cardinality estimation. In this paper, we investigate probabilistic counters from the perspective of preserving privacy. We use the standard, rigid differential privacy notion. The intuition is that the probabilistic counters do not reveal too much information about individuals but provide only general information about the population. Therefore, they can be used safely without violating the privacy of individuals. However, it turned out, that providing a precise, formal analysis of the privacy parameters of probabilistic counters is surprisingly difficult and needs advanced techniques and a very careful approach. We demonstrate that probabilistic counters can be used as a privacy protection mechanism without extra randomization. Namely, the inherent randomization from the protocol is sufficient for protecting privacy, even if the probabilistic counter is used multiple times. In particular, we present a specific privacy-preserving data aggregation protocol based on Morris Counter and MaxGeo Counter. Some of the presented results are devoted to counters that have not been investigated so far from the perspective of privacy protection. Another part is an improvement of previous results. We show how our results can be used to perform distributed surveys and compare the properties of counter-based solutions and a standard Laplace method.
Section: Combinatorics
12. K-promotion on m-packed labelings of posets
Schutzenberger's promotion operator, pro, is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But pro was subsequently shown to have interesting properties when applied to natural labelings of other posets. Pechenik defined a K-theoretic version of promotion, pro_K, on m-packed labelings of tableaux. The operator pro was then extended to increasing labelings of other posets. The purpose of the current work is to show that the original action of pro_K on m-packed labelings yields interesting results when applied to partially ordered sets in general, and to rooted trees in particular. We show that under certain conditions, the sizes of the orbits and order of pro_K exhibit nice divisibility properties. We also completely determine, for certain values of m, the orbit sizes for the action on various types of rooted trees such as extended stars, combs, zippers, and a type of three-leaved tree.
Section: Combinatorics