Planar Bipartite Graphs Research Articles

Artificial Intelligence Approaches to Build Ticket to Ride Maps

Fun, as a game trait, is challenging to evaluate. Previous research explores game arc and game refinement to improve the quality of games. Fun, for some players, is having an even chance to win while executing their strategy. To explore this, we build boards for the game Ticket to Ride while optimizing for a given win rate between four AI agents. These agents execute popular strategies human players use: one-step thinking, long route exploitation, route focus, and destination hungry strategies. We create the underlying graph of a map by connecting several planar bipartite graphs. To build the map, we use a multiple phase design, with each phase implementing several simplified Monte Carlo Tree Search components. Within a phase, the components communicate with each other passively. The experiments show that the proposed approach results in improvements over randomly generated graphs and maps.

Total 2-Rainbow Domination in Graphs

A total k-rainbow dominating function on a graph G=(V,E) is a function f:V(G)→2{1,2,…,k} such that (i) ∪u∈N(v)f(u)={1,2,…,k} for every vertex v with f(v)=∅, (ii) ∪u∈N(v)f(u)≠∅ for f(v)≠∅. The weight of a total 2-rainbow dominating function is denoted by ω(f)=∑v∈V(G)|f(v)|. The total k-rainbow domination number of G is the minimum weight of a total k-rainbow dominating function of G. The minimum total 2-rainbow domination problem (MT2RDP) is to find the total 2-rainbow domination number of the input graph. In this paper, we study the total 2-rainbow domination number of graphs. We prove that the MT2RDP is NP-complete for planar bipartite graphs, chordal bipartite graphs, undirected path graphs and split graphs. Then, a linear-time algorithm is proposed for computing the total k-rainbow domination number of trees. Finally, we study the difference in complexity between MT2RDP and the minimum 2-rainbow domination problem.

Excluded checkerboard colourable ribbon graph minors

Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.

Families of convex tilings

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.

The cubical matching complex revisited

Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the disjoint union of two or more collapsible complexes. We also prove that all links in these complexes are suspensions up to homotopy. Furthermore, we extend the definition of a cubical matching complex to planar graphs that are not necessarily bipartite, and show that these complexes are either contractible or a disjoint union of contractible complexes. For a simple connected region that can be tiled with dominoes (2 × 1 and 1 × 2) and 2 × 2 squares, let fi denote the number of tilings with exactly i squares. We prove that f0 − f1 + f2 − f3 + ⋯ = 1 (established by Ehrenborg) is the only linear relation for the numbers fi.

The complexity of the unit stop number problem and its implications to other related problems

The Stop Number Problem arises in the management of a dial-a-ride system served by a fleet of autonomous electric vehicles. In such a system, clients request for a ride from an origin station to a destination station, and a fleet of capacitated vehicles must satisfy all requests. The goal is to minimize the number of pick-up/drop-off operations. In this paper we focus on a constrained unitary variant, called Unit Stop Number Problem, where each client requests for a single seat in the vehicles. This variant was recently conjectured to be NP-Hard. In this regard, we show how this problem relates to other problems known in the literature in order to derive some polynomial-time solvable variants. Moreover, we provide a positive answer to the conjecture by showing that the problem is NP-Hard for any fixed capacity greater than or equal to 2, even for the case where the graph of requests is restricted to the class of planar bipartite graphs. Our proof of NP-Hardness also improves the complexity results known in the literature for the related problems identified.

Continuous forcing spectrum of regular hexagonal polyhexes

For any perfect matching M of a graph G, the forcing number (resp. anti-forcing number) of M is the smallest cardinality of an edge subset S⊆M (resp. S⊆E(G)∖M) such that the graph G−V(S) (resp. G−S) has a unique perfect matching. The forcing spectrum of G is the set of forcing numbers of all perfect matchings of G. Afshani et al. [2] proved that any finite set of positive integers can be the forcing spectrum of a planar bipartite graph. A polyhex is a 2-connected plane bipartite graph whose interior faces are regular hexagons. In this paper, we give the minimum forcing number and anti-forcing number of a polyhex with a 3-divisible perfect matching. Further, we prove that the forcing spectrum of any regular hexagonal polyhex is continuous, i.e. an integer interval, by applying some changes on prolate triangle polyhexes to obtain a sequence of perfect matchings of the graph.

Domination and dominator colorings in planar graphs with small diameter

A dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class, where a vertex dominates itself and all vertices adjacent to it. The dominator chromatic number of G is the minimum number of colors among all dominator coloring of G. A total dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class other than its own. The total dominator chromatic number of G is the minimum number of colors among all total dominator coloring of G. In this paper, we present bounds on the dominator chromatic number and total dominator chromatic number of a planar graphs with small diameter. In particular we show the dominator chromatic number of a planar graph of diameter 2 is at most 5. We also present results for the special cases of outerplanar graphs and bipartite planar graphs.

Star colouring of bounded degree graphs and regular graphs

A k-star colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that f(u)≠f(v) for every edge uv of G, and every bicoloured connected subgraph of G is a star. The star chromatic number of G, χs(G), is the least integer k such that G is k-star colourable. We prove that χs(G)≥⌈(d+4)/2⌉ for every d-regular graph G with d≥3. We reveal the structure and properties of even-degree regular graphs G that attain this lower bound. The structure of such graphs G is linked with a certain type of Eulerian orientations of G. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for p≥2, a 2p-regular graph G is (p+2)-star colourable only if n:=|V(G)| is divisible by (p+1)(p+2). For each p≥2 and n divisible by (p+1)(p+2), we construct a 2p-regular Hamiltonian graph on n vertices which is (p+2)-star colourable.The problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. We prove that 3-Star Colourability is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For k≥3, k-Star Colourability of bipartite graphs of maximum degree k is NP-complete, and does not even admit a 2o(n)-time algorithm unless ETH fails.

Open-independent, open-locating-dominating sets: structural aspects of some classes of graphs

Let $G=(V(G),E(G))$ be a finite simple undirected graph with vertex set $V(G)$, edge set $E(G)$ and vertex subset $S\subseteq V(G)$. $S$ is termed \emph{open-dominating} if every vertex of $G$ has at least one neighbor in $S$, and \emph{open-independent, open-locating-dominating} (an $OLD_{oind}$-set for short) if no two vertices in $G$ have the same set of neighbors in $S$, and each vertex in $S$ is open-dominated exactly once by $S$. The problem of deciding whether or not $G$ has an $OLD_{oind}$-set has important applications that have been reported elsewhere. As the problem is known to be $\mathcal{NP}$-complete, it appears to be notoriously difficult as we show that its complexity remains the same even for just planar bipartite graphs of maximum degree five and girth six, and also for planar subcubic graphs of girth nine. Also, we present characterizations of both $P_4$-tidy graphs and the complementary prisms of cographs that have an $OLD_{oind}$-set.

A polynomial version of Cereceda's conjecture

Let k and d be positive integers such that k≥d+2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper colouring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length.The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d+2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n2). So far, there is no proof of a polynomial upper bound on the diameter, even for d=2.In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(nd+1) for k≥d+2. Moreover, we prove that if k≥32(d+1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k≥2d+1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.

The Outer-Paired Domination of Graphs

In this paper, we introduce a new variant of domination called the outer-paired domination. For a graph [Formula: see text], an outer-paired dominating set [Formula: see text] is a dominating set of [Formula: see text] such that the induced subgraph of [Formula: see text] contains a perfect matching. The outer-paired domination number of a graph [Formula: see text] is the cardinality of a minimum outer-paired dominating set of [Formula: see text]. We show that finding the outer-paired domination number of a graph [Formula: see text] is NP-hard on bipartite graphs, chordal graphs, and planar graphs. We also propose a linear-time algorithm for solving the outer-paired domination problem on trees.

Degree powers in Ks,t-minor free graphs

Let G be a simple graph with the degree sequence (d1,d2,…,dn). Given a positive integer p, denote the degree power of G by ep(G)=∑i=1ndip. In this paper, we study the following problem: what is the maximum value of ep(G) where G is a Ks,t-minor free graph of order n? By using majorization, we solve the problem for K2,3-minor free and K3,3-minor free graphs and characterize the extremal graphs. The results generalize the results of outerplanar [28] and planar graphs [18,28]. Ding, Johnson and Seymour [7] proved the maximum value of e1(G) when G is K1,t-minor free. We generalize the result by showing the maximum value of ep(G) for p≥2 among all K1,t-minor free graphs and characterizing the extremal graph. Further, we determine the maximum degree power for bipartite planar graphs and the minimum degree power for non-planar graphs, and the corresponding extremal graphs are also determined. In the end, we conclude this paper with an open problem.

Further Results on Total Edge‐Vertex Domination

Total edge‐vertex domination is a new total domination‐type parameter. In this paper, the author shows that determining the total edge‐vertex domination number in bipartite planar graphs is NP‐complete. Also, the author obtains a structural relation between total domination number and total edge‐vertex domination number and characterizes the trees whose total edge‐vertex domination number is equal to total domination number.

Finding Dominating Induced Matchings in $P_{10}$-Free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge subset $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m. in $G$. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but was solved in linear time for $P_7$-free graphs and in polynomial time for $P_8$-free graphs. In this paper, we solve it in polynomial time for $P_9$-free graphs.

On the Parameterized Complexity Of the Acyclic Matching Problem

A matching is a set of edges in a graph with no common endpoint. A matching $M$ is called acyclic if the induced subgraph on the endpoints of the edges in $M$ is acyclic. Given a graph $G$ and an integer $k$, Acyclic Matching Problem seeks for an acyclic matching of size $k$ in $G$. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter $k$. On the other hand, the problem is fixed parameter tractable with respect to $k$, for line graphs, $C_4$-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

Density of C−4-critical signed graphs

A signed bipartite (simple) graph (G,σ) is said to be C−4-critical if it admits no homomorphism to C−4 (a negative 4-cycle) but each of its proper subgraphs does. To motivate the study of C−4-critical signed graphs, we show that the notion of 4-coloring of graphs and signed graphs is captured, through simple graph operations, by the notion of homomorphism to C−4. In particular, the 4-color theorem is equivalent to: Given a planar graph G, the signed bipartite graph obtained from G by replacing each edge with a negative path of length 2 maps to C−4.We prove that, except for one particular signed bipartite graph on 7 vertices and 9 edges, any C−4-critical signed graph on n vertices must have at least ⌈4n3⌉ edges. Moreover, we show that for each value of n≥9 there exists a C−4-critical signed graph on n vertices with either ⌈4n3⌉ or ⌈4n3⌉+1 many edges.As an application, we conclude that all signed bipartite planar graphs of negative girth at least 8 map to C−4. Furthermore, we show that there exists an example of a signed bipartite planar graph of girth 6 which does not map to C−4, showing 8 is the best possible and disproving a conjecture of Naserasr, Rollova and Sopena.

The minimality of the Georges–Kelmans graph

In 1971, Tutte wrote in an article that it is tempting to conjecture that every 3-connected bipartite cubic graph is hamiltonian. Motivated by this remark, Horton constructed a counterexample on 96 96 vertices. In a sequence of articles by different authors several smaller counterexamples were presented. The smallest of these graphs is a graph on 50 vertices which was discovered independently by Georges and Kelmans. In this article we show that there is no smaller counterexample. As all non-hamiltonian 3-connected bipartite cubic graphs in the literature have cyclic 4-cuts—even if they have girth 6—it is natural to ask whether this is a necessary prerequisite. In this article we answer this question in the negative and give a construction of an infinite family of non-hamiltonian cyclically 5-connected bipartite cubic graphs. In 1969 Barnette gave a weaker version of the conjecture stating that 3-connected planar bipartite cubic graphs are hamiltonian. We show that Barnette’s conjecture is true up to at least 90 vertices. We also report that a search of small non-hamiltonian 3-connected bipartite cubic graphs did not find any with genus less than 4.

Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk

Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. 30(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) 92(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. 25(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.