# vol. 24, no 2

### 1. Further results on Hendry's Conjecture

Recently, a conjecture due to Hendry was disproved which stated that every Hamiltonian chordal graph is cycle extendible. Here we further explore the conjecture, showing that it fails to hold even when a number of extra conditions are imposed. In particular, we show that Hendry's Conjecture fails for strongly chordal graphs, graphs with high connectivity, and if we relax the definition of "cycle extendible" considerably. We also consider the original conjecture from a subtree intersection model point of view, showing that a result of Abuieda et al is nearly best possible.
Section: Graph Theory

### 2. On the domination number of $t$-constrained de Bruijn graphs

Motivated by the work on the domination number of directed de Bruijn graphs and some of its generalizations, in this paper we introduce a natural generalization of de Bruijn graphs (directed and undirected), namely $t$-constrained de Bruijn graphs, where $t$ is a positive integer, and then study the domination number of these graphs. Within the definition of $t$-constrained de Bruijn graphs, de Bruijn and Kautz graphs correspond to 1-constrained and 2-constrained de Bruijn graphs, respectively. This generalization inherits many structural properties of de Bruijn graphs and may have similar applications in interconnection networks or bioinformatics. We establish upper and lower bounds for the domination number on $t$-constrained de Bruijn graphs both in the directed and in the undirected case. These bounds are often very close and in some cases we are able to find the exact value.
Section: Graph Theory

### 3. On the monophonic rank of a graph

A set of vertices $S$ of a graph $G$ is {\em monophonically convex} if every induced path joining two vertices of $S$ is contained in $S$. The {\em monophonic convex hull of $S$}, $\langle S \rangle$, is the smallest monophonically convex set containing $S$. A set $S$ is {\em monophonic convexly independent} if $v \not\in \langle S - \{v\} \rangle$ for every $v \in S$. The {\em monophonic rank} of $G$ is the size of the largest monophonic convexly independent set of $G$. We present a characterization of the monophonic convexly independent sets. Using this result, we show how to determine the monophonic rank of graph classes like bipartite, cactus, triangle-free, and line graphs in polynomial time. Furthermore, we show that this parameter can computed in polynomial time for $1$-starlike graphs, i.e., for split graphs, and that its determination is $\NP$-complete for $k$-starlike graphs for any fixed $k \ge 2$, a subclass of chordal graphs. We also consider this problem on the graphs whose intersection graph of the maximal prime subgraphs is a tree.
Section: Graph Theory

### 4. Extremal problems of double stars

In a generalized Turán problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.
Section: Graph Theory

### 5. The 2-colouring problem for $(m,n)$-mixed graphs with switching is polynomial

A mixed graph is a set of vertices together with an edge set and an arc set. An $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one of $m$ colours, and whose arcs are each assigned one of $n$ colours. A \emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc colours, and the arc directions of edges and arcs incident with $v$. The group of all allowed switches is $\Gamma$. Let $k \geq 1$ be a fixed integer and $\Gamma$ a fixed permutation group. We consider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if there a sequence of switches at vertices of $G$ with respect to $\Gamma$ so that the resulting $(m,n)$-mixed graph admits a homomorphism to an $(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem can be solved in polynomial time for $k \leq 2$, and is NP-hard for $k \geq 3$. This provides a step towards a general dichotomy theorem for the $\Gamma$-switchable homomorphism decision problem.
Section: Graph Theory

### 6. Improved product structure for graphs on surfaces

Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes K_{\max\{2g,3\}}$. We improve this result by replacing "4" by "3" and with $H$ planar. We in fact prove a more general result in terms of so-called framed graphs. This implies that every $(g,d)$-map graph is contained in $H \boxtimes P\boxtimes K_\ell$, for some planar graph $H$ with treewidth $3$, where $\ell=\max\{2g\lfloor \frac{d}{2} \rfloor,d+3\lfloor\frac{d}{2}\rfloor-3\}$. It also implies that every $(g,1)$-planar graph (that is, graphs that can be drawn in a surface of Euler genus $g$ with at most one crossing per edge) is contained in $H\boxtimes P\boxtimes K_{\max\{4g,7\}}$, for some planar graph $H$ with treewidth $3$.
Section: Graph Theory

### 7. On Dualization over Distributive Lattices

Given a partially order set (poset) $P$, and a pair of families of ideals $\mathcal{I}$ and filters $\mathcal{F}$ in $P$ such that each pair $(I,F)\in \mathcal{I}\times\mathcal{F}$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\mathcal{F}$ and does not contain any member of $\mathcal{I}$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains $\mathcal{A},\mathcal{B}\subseteq L$ such that no $a\in\mathcal{A}$ is dominated by any $b\in\mathcal{B}$, whether $\mathcal{A}$ and $\mathcal{B}$ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of $P$, $\mathcal{A}$ and $\mathcal{B}$, thus answering an open question in Babin and Kuznetsov (2017). As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.
Section: Discrete Algorithms

### 8. Approximability results for the $p$-centdian and the converse centdian problems

Given an undirected graph $G=(V,E)$ with a nonnegative edge length function and an integer $p$, $0 < p < |V|$, the $p$-centdian problem is to find $p$ vertices (called the {\it centdian set}) of $V$ such that the {\it eccentricity} plus {\it median-distance} is minimized, in which the {\it eccentricity} is the maximum (length) distance of all vertices to their nearest {\it centdian set} and the {\it median-distance} is the total (length) distance of all vertices to their nearest {\it centdian set}. The {\it eccentricity} plus {\it median-distance} is called the {\it centdian-distance}. The purpose of the $p$-centdian problem is to find $p$ open facilities (servers) which satisfy the quality-of-service of the minimum total distance ({\it median-distance}) and the maximum distance ({\it eccentricity}) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the {\it centdian-distance} and the objective function is to minimize the cardinality of the {\it centdian set}, this problem is called the converse centdian problem. In this paper, we prove the $p$-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the $p$-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.
Section: Discrete Algorithms

### 9. Token Swapping on Trees

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.
Section: Discrete Algorithms

### 10. Proximity, remoteness and maximum degree in graphs

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.
Section: Graph Theory

### 11. A heuristic technique for decomposing multisets of non-negative integers according to the Minkowski sum

We study the following problem. Given a multiset $M$ of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to $M$. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds independently of the magnitude of numbers belonging to the multisets. Our heuristic can be used also for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab.
Section: Analysis of Algorithms

### 12. Series acceleration formulas obtained from experimentally discovered hypergeometric recursions

In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating function identities used to obtain series accelerations for values of Dirichlet's $\beta$ function, via the Markov--Wilf--Zeilberger method. Inspired by these past results, together with related results introduced by Chu et al., we introduce a variety of hypergeometric recurrences. We prove these recurrences using the WZ method, and we apply these recurrences to obtain series acceleration identities. We introduce a family of summations generalizing a Ramanujan-type series for $\frac{1}{\pi^2}$ due to Guillera, and a family of summations generalizing an accelerated series for Catalan's constant due to Lupa\c{s}, and many related results.
Section: Analysis of Algorithms