vol. 20 no. 2

1. Tropical Vertex-Disjoint Cycles of a Vertex-Colored Digraph: Barter Exchange with Multiple Items Per Agent

Highley, Timothy ; Le, Hoang.
In a barter exchange market, agents bring items and seek to exchange their items with one another. Agents may agree to a k-way exchange involving a cycle of k agents. A barter exchange market can be represented by a digraph where the vertices represent items and the edges out of a vertex indicate the items that an agent is willing to accept in exchange for that item. It is known that the problem of finding a set of vertex-disjoint cycles with the maximum total number of vertices (MAX-SIZE-EXCHANGE) can be solved in polynomial time. We consider a barter exchange where each agent may bring multiple items, and items of the same agent are represented by vertices with the same color. A set of cycles is said to be tropical if for every color there is a cycle that contains a vertex of that color. We show that the problem of determining whether there exists a tropical set of vertex-disjoint cycles in a digraph (TROPICAL-EXCHANGE) is NP-complete and APX-hard. This is equivalent to determining […]
Section: Analysis of Algorithms

2. On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

Rajaati, M. ; Hooshmandasl, M. R. ; Dinneen, M. J. ; Shakiba, A..
A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, […]
Section: Graph Theory

3. Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers

Proctor, Robert A. ; Willis, Matthew J..
This is the first of three papers that develop structures which are counted by a "parabolic" generalization of Catalan numbers. Fix a subset R of {1,..,n-1}. Consider the ordered partitions of {1,..,n} whose block sizes are determined by R. These are the "inverses" of (parabolic) multipermutations whose multiplicities are determined by R. The standard forms of the ordered partitions are refered to as "R-permutations". The notion of 312-avoidance is extended from permutations to R-permutations. Let lambda be a partition of N such that the set of column lengths in its shape is R or R union {n}. Fix an R-permutation pi. The type A Demazure character (key polynomial) in x_1, .., x_n that is indexed by lambda and pi can be described as the sum of the weight monomials for some of the semistandard Young tableau of shape lambda that are used to describe the Schur function indexed by lambda. Descriptions of these "Demazure" tableaux developed by the […]
Section: Combinatorics

4. Semitotal domination in trees

Wei, Zhuang ; Guoliang, Hao.
In this paper, we study a parameter that is squeezed between arguably the two important domination parameters, namely the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$. A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in S is within distance $2$ of another vertex of $S$. The semitotal domination number, $\gamma_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. We observe that $\gamma(G)\leq \gamma_{t2}(G)\leq \gamma_t(G)$. In this paper, we give a lower bound for the semitotal domination number of trees and we characterize the extremal trees. In addition, we characterize trees with equal domination and semitotal domination numbers.
Section: Graph Theory

5. Fast strategies in biased Maker--Breaker games

Mikalački, Mirjana ; Stojaković, Miloš.
We study the biased $(1:b)$ Maker--Breaker positional games, played on the edge set of the complete graph on $n$ vertices, $K_n$. Given Breaker's bias $b$, possibly depending on $n$, we determine the bounds for the minimal number of moves, depending on $b$, in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game.
Section: Graph Theory

6. On locally irregular decompositions and the 1-2 Conjecture in digraphs

Baudon , Olivier ; Bensmail , Julien ; Przybyło , Jakub ; Woźniak , Mariusz.
The 1-2 Conjecture raised by Przybylo and Wozniak in 2010 asserts that every undirected graph admits a 2-total-weighting such that the sums of weights "incident" to the vertices yield a proper vertex-colouring. Following several recent works bringing related problems and notions (such as the well-known 1-2-3 Conjecture, and the notion of locally irregular decompositions) to digraphs, we here introduce and study several variants of the 1-2 Conjecture for digraphs. For every such variant, we raise conjectures concerning the number of weights necessary to obtain a desired total-weighting in any digraph. We verify some of these conjectures, while we obtain close results towards the ones that are still open.
Section: Graph Theory

7. Steiner Distance in Product Networks

Mao, Yaping ; Cheng, Eddie ; Wang, Zhao.
For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In this paper, we investigate the Steiner distance and Steiner $k$-diameter of Cartesian and lexicographical product graphs. Also, we study the Steiner $k$-diameter of some networks.
Section: Graph Theory

8. Summation formulas for Fox-Wright function

Wei, Chuanan ; Liu, Lily Li ; Gong, Dianxuan.
By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.
Section: Combinatorics

9. Parameterized Power Vertex Cover

Angel, Eric ; Bampis, Evripidis ; Escoffier, Bruno ; Lampis, Michael.
We study a recently introduced generalization of the Vertex Cover (VC) problem, called Power Vertex Cover (PVC). In this problem, each edge of the input graph is supplied with a positive integer demand. A solution is an assignment of (power) values to the vertices, so that for each edge one of its endpoints has value as high as the demand, and the total sum of power values assigned is minimized. We investigate how this generalization affects the parameterized complexity of Vertex Cover. On the positive side, when parameterized by the value of the optimal P, we give an O*(1.274^P)-time branching algorithm (O* is used to hide factors polynomial in the input size), and also an O*(1.325^P)-time algorithm for the more general asymmetric case of the problem, where the demand of each edge may differ for its two endpoints. When the parameter is the number of vertices k that receive positive value, we give O*(1.619^k) and O*(k^k)-time algorithms for the symmetric and asymmetric cases […]
Section: Discrete Algorithms

10. The 26 Wilf-equivalence classes of length five quasi-consecutive patterns

Chen, Evan ; Narayanan, Shyam.
We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.
Section: Combinatorics

11. IMP with exceptions over decorated logic

Ekici , Burak.
In this paper, we facilitate the reasoning about impure programming languages, by annotating terms with “decorations”that describe what computational (side) effect evaluation of a term may involve. In a point-free categorical language,called the “decorated logic”, we formalize the mutable state and the exception effects first separately, exploiting anice duality between them, and then combined. The combined decorated logic is used as the target language forthe denotational semantics of the IMP+Exc imperative programming language, and allows us to prove equivalencesbetween programs written in IMP+Exc. The combined logic is encoded in Coq, and this encoding is used to certifysome program equivalence proofs.
Section: Automata, Logic and Semantics

12. General bounds on limited broadcast domination

Cáceres, José ; Hernando, Carmen ; Mora, Mercè ; Pelayo, Ignacio M. ; Puertas, María Luz.
Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical domination. In this limited version, the broadcast function is upper bounded by an integer $k$ and the minimum cost of such function is the dominating $k$-broadcast number. Our main result is a unified upper bound on this parameter for any value of $k$ in general graphs, in terms of both $k$ and the order of the graph. We also study the computational complexity of the associated decision problem.
Section: Graph Theory

13. A Note on Flips in Diagonal Rectangulations

Cardinal, Jean ; Sacristán, Vera ; Silveira, Rodrigo I..
Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal with local changes involving a single edge of a rectangulation, referred to as flips, edge rotations, or edge pivoting. Such operations induce a graph on equivalence classes of rectangulations, related to so-called flip graphs on triangulations and other families of geometric partitions. In this note, we consider a family of flip operations on the equivalence classes of diagonal rectangulations, and their interpretation as transpositions in the associated Baxter permutations, avoiding the vincular patterns { 3{14}2, 2{41}3 }. This complements results from Law and Reading (JCTA, 2012) and provides a complete characterization of flip operations on diagonal rectangulations, in both geometric and combinatorial terms.
Section: Combinatorics