 |
Discrete Mathematics & Theoretical Computer Science |
vol. 26:2
1. Composing dynamic programming tree-decomposition-based algorithms
Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, \break $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[S_i] \in \mathcal{H}_i$, $G[S_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We apply this approach to known problems. For well-studied problems, like VERTEX COVER and GRAPH $q$-COLORING, we obtain running times that are comparable to those of the best known problem-specific algorithms. For an exotic problem from bioinformatics, called DISPLAYGRAPH, this […]
Section:
Discrete Algorithms
2. Bijective proof of a conjecture on unit interval posets
In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by Gélinas, Segovia and Thomas using induction. In this short note, we provide a bijective proof of the same conjecture with a reformulation of the zeta map using left-aligned colored trees, first proposed in the study of parabolic Tamari lattices.
Section:
Combinatorics
3. On the $\operatorname{rix}$ statistic and valley-hopping
This paper studies the relationship between the modified Foata$\unicode{x2013}$Strehl action (a.k.a. valley-hopping)$\unicode{x2014}$a group action on permutations used to demonstrate the $\gamma$-positivity of the Eulerian polynomials$\unicode{x2014}$and the number of rixed points $\operatorname{rix}$$\unicode{x2014}$a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the $\operatorname{rix}$ statistic is homomesic under valley-hopping. We also demonstrate that a bijection $\Phi$ introduced by Lin and Zeng in the study of the $\operatorname{rix}$ statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points $\operatorname{fix}$ is homomesic under cyclic valley-hopping.
Section:
Combinatorics
4. Weakly toll convexity and proper interval graphs
A walk $u_0u_1 \ldots u_{k-1}u_k$ is a \textit{weakly toll walk} if $u_0u_i \in E(G)$ implies $u_i = u_1$ and $u_ju_k\in E(G)$ implies $u_j=u_{k-1}$. A set $S$ of vertices of $G$ is {\it weakly toll convex} if for any two non-adjacent vertices $x,y \in S$ any vertex in a weakly toll walk between $x$ and $y$ is also in $S$. The {\em weakly toll convexity} is the graph convexity space defined over weakly toll convex sets. Many studies are devoted to determine if a graph equipped with a convexity space is a {\em convex geometry}. An \emph{extreme vertex} is an element $x$ of a convex set $S$ such that the set $S\backslash\{x\}$ is also convex. A graph convexity space is said to be a convex geometry if it satisfies the Minkowski-Krein-Milman property, which states that every convex set is the convex hull of its extreme vertices. It is known that chordal, Ptolemaic, weakly polarizable, and interval graphs can be characterized as convex geometries with respect to the monophonic, geodesic, $m^3$, and toll convexities, respectively. Other important classes of graphs can also be characterized in this way. In this paper, we prove that a graph is a convex geometry with respect to the weakly toll convexity if and only if it is a proper interval graph. Furthermore, some well-known graph invariants are studied with respect to the weakly toll convexity.
Section:
Graph Theory
5. Extending partial edge colorings of iterated cartesian products of cycles and paths
We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if $G=C^d_{2k}$ is the $d$th power of the cartesian product of the even cycle $C_{2k}$ with itself, and at most $2d-1$ edges of $G$ are precolored, then there is a proper $2d$-edge coloring of $G$ that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least $4$ from each other. For odd cycles of length at least $5$, we prove that if $G=C^d_{2k+1}$ is the $d$th power of the cartesian product of the odd cycle $C_{2k+1}$ with itself ($k\geq2$), and at most $2d$ edges of $G$ are precolored, then there is a proper $(2d+1)$-edge coloring of $G$ that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410--444].
Section:
Graph Theory
6. On harmonious coloring of hypergraphs
A harmonious coloring of a $k$-uniform hypergraph $H$ is a vertex coloring such that no two vertices in the same edge have the same color, and each $k$-element subset of colors appears on at most one edge. The harmonious number $h(H)$ is the least number of colors needed for such a coloring. The paper contains a new proof of the upper bound $h(H)=O(\sqrt[k]{k!m})$ on the harmonious number of hypergraphs of maximum degree $\Delta$ with $m$ edges. We use the local cut lemma of A. Bernshteyn.
Section:
Graph Theory
7. A new sufficient condition for a 2-strong digraph to be Hamiltonian
In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.
Section:
Graph Theory
8. A note on removable edges in near-bricks
An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$ different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable edges, where $\Delta$ is the maximum degree of $G$. In this paper, we generalize the result to irreducible near-bricks, where a graph is irreducible if it contains no single ear of length three or more.
Section:
Graph Theory
9. Coloring Groups
We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.
Section:
Combinatorics
10. Representing Matroids over the Reals is $\exists \mathbb R$-complete
A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| < |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $\varphi(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.
Section:
Discrete Algorithms
11. On the Complexity of Target Set Selection in Simple Geometric Networks
We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.
Section:
Discrete Algorithms
12. Line game-perfect graphs
The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $\chi_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, \[\chi_{[X,Y]}'(H)=\omega(L(H)),\] where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.
Section:
Graph Theory
13. Leanness Computation: Small Values and Special Graph Classes
Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.
Section:
Graph Theory
14. On polynomials associated to Voronoi diagrams of point sets and crossing numbers
Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.
Section:
Combinatorics