Planar Bipartite Graphs Research Articles

Induced forests in bipartite planar graphs

Akiyama and Watanabe conjectured that every simple planar bipartite graph on $n$ vertices contains an induced forest on at least $5n/8$ vertices. We apply the discharging method to show that every simple bipartite planar graph on $n$ vertices contains an induced forest on at least $\lceil (4n+3)/7 \rceil$ vertices.

Complexity of rainbow vertex connectivity problems for restricted graph classes

A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the complexity of deciding if a given vertex-colored graph is rainbow or strongly rainbow vertex connected. We call these problems Rainbow Vertex Connectivity and Strong Rainbow Vertex Connectivity, respectively. We prove both problems remain NP-complete on very restricted graph classes including bipartite planar graphs of maximum degree 3, interval graphs, and k-regular graphs for k≥3. We settle precisely the complexity of both problems from the viewpoint of two width parameters: pathwidth and tree-depth. More precisely, we show both problems remain NP-complete for bounded pathwidth graphs, while being fixed-parameter tractable parameterized by tree-depth. Moreover, we show both problems are solvable in polynomial time for block graphs, while Strong Rainbow Vertex Connectivity is tractable for cactus graphs and split graphs.

Large Induced Forests in Graphs

AbstractIn this article, we prove three theorems. The first is that every connected graph of order n and size m has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to a clique of order either 4 or 5. This improves the previous lower bound of Alon–Kahn–Seymour for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Albertson and Berman that every planar graph of order n has an induced forest of order at least . The second is that every connected triangle‐free graph of order n and size m has an induced forest of order at least . This bound is sharp by the cube and the Wagner graph. It also improves the previous lower bound of Alon–Mubayi–Thomas for , and implies that such a graph has an induced forest of order at least for . This latter result relates to the conjecture of Akiyama and Watanabe that every bipartite planar graph of order n has an induced forest of order at least . The third is that every connected planar graph of order n and size m with girth at least 5 has an induced forest of order at least with equality if and only if such a graph is obtained from a tree by expanding every vertex to one of five specific graphs. This implies that such a graph has an induced forest of order at least , where was conjectured to be the best lower bound by Kowalik, Lužar, and Škrekovski.

Generalizing the Divisibility Property of Rectangle Domino Tilings

We introduce a class of graphs called compound graphs, which are constructed out of copies of a planar bipartite base graph, and explore the number of perfect matchings of compound graphs. The main result is that the number of matchings of every compound graph is divisible by the number of matchings of its base graph. Our approach is to use Kasteleyn's theorem to prove a key lemma, from which the divisibility theorem follows combinatorially. This theorem is then applied to provide a proof of Problem 21 of Propp's Enumeration of Matchings, a divisibility property of rectangles. Finally, we present a new proof, in the same spirit, of Ciucu's factorization theorem.

Chip Removal for Computing the Number of Perfect Matchings

We consider a transformation of a graph G that replaces an induced subgraph H of arbitrary size by a small new subgraph h. We choose h in such a way that the equality M(G) = xM(G′) holds (where G′ is the new graph and the factor x depends on the numbers of matchings of H and its subgraphs). We describe how one can construct h when G is a plane graph and H is a bipartite graph (with some restriction on the coloring of the vertices connecting it with the other part of the graph G). For a plane bipartite graph H with a small number of such vertices, we prove that the equality holds for an arbitrary graph G.

The VC-dimension of graphs with respect to [formula omitted]-connected subgraphs

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its k-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is NP-complete and that it remains NP-complete for split graphs and for some subclasses of planar bipartite graphs in the cases k=1 and k=2. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width.

Finding Dominating Induced Matchings in $$P_8$$ P 8 -Free Graphs in Polynomial Time

Let $$G=(V,E)$$G=(V,E) be a finite undirected graph. An edge set $$E' \subseteq E$$E?⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of $$E'$$E?. The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is $${\mathbb {NP}}$$NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for $$P_7$$P7-free graphs. However, its complexity was open for $$P_k$$Pk-free graphs for any $$k \ge 8$$k?8; $$P_k$$Pk denotes the chordless path with k vertices and $$k-1$$k-1 edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $$P_8$$P8-free graphs.

Weighted restrained domination in subclasses of planar graphs

An undirected simple and connected graph is denoted by G(V,E), where V and E are the vertex-set and the edge-set of G, respectively. For any subset S of V,〈S〉 denotes the subgraph of G induced by S. A subset Q of V is a restrained dominating set of G if ⋃u∈QN[u]=V and no isolated vertices appear in 〈V−Q〉, where N(x)={u|(u,x)∈E} and N[x]=N(x)∪{x}, for all x∈V. This paper studies the Weighted Restrained Domination problem (the WRD problem) on graphs G(V,E,w), where w is a function giving a positive weight w(v) to each vertex v. The problem is to find a restrained dominating set D of the given graph G(V,E,w) and the objective is to minimize w(D)=∑v∈Dw(v). When w(v)=1, for all v∈V, the WRD problem is abbreviated as the RD problem. The first result shows that the WRD problem on cactus graphs can be solved in linear time. The second result proves that the RD problem on planar bipartite graphs remains NP-hard.

The Degree/Diameter Problem in Maximal Planar Bipartite graphs

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

Global defensive sets in graphs

In the paper we study a new problem of finding a minimum global defensive set in a graph which is a generalization of the global alliance problem. For a given graph G and a subset S of a vertex set of G, we define for every subset X of S the predicate SEC(X)=true if and only if |N[X]∩S|≥|N[X]∖S| holds, where N[X] is a closed neighbourhood of X in graph G. A set S is a defensive alliance if and only if for each vertex v∈S we have SEC({v})=true. If S is also a dominating set of G (i.e., N[S]=V(G)), we say that S is a global defensive alliance.We introduce the concept of defensive sets in graph G as follows: set S is a defensive set in G if and only if for each vertex v∈S we have SEC({v})=true or there exists a neighbour u of v such that u∈S and SEC({v,u})=true. Similarly, if S is also a dominating set of G, we say that S is a global defensive set. We also study the problems of total dominating alliances (total alliances) and total dominating defensive sets (total defensive sets), i.e., S is a dominating set and the induced graph G[S] has no isolated vertices.In the paper we proved the NP-completeness for planar bipartite subcubic graphs of the decision versions of the following minimalization problems: a global and total alliance, a global and total defensive set. We proposed polynomial time algorithms solving in trees the problem of finding the minimum total and global defensive set and the total alliance. We obtained the lower bound on the minimum size of a global defensive set in arbitrary graphs and trees.

A minimax result for perfect matchings of a polyomino graph

Let G be a plane bipartite graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M such that S is not contained by other perfect matchings of G. The minimum cardinality of forcing sets of M is called the forcing number of M, denoted by f(G,M). Pachter and Kim established a minimax result: for any perfect matching M of G, f(G,M) is equal to the maximum number of disjoint M-alternating cycles in G. For a polyomino graph H, we show that for every perfect matching M of H with the maximum or second maximum forcing number, f(H,M) is equal to the maximum number of disjoint M-alternating squares in H. This minimax result does not hold in general for other perfect matchings of H with smaller forcing number.

Adjacent vertex distinguishing index of bipartite planar graphs

A proper edge coloring of a graph $G$ is called adjacent vertex distinguishing, if any pair of adjacent vertices meet distinct color sets. The adjacent vertex distinguishing index of $G$, denoted by $\chi_a(G)$, is the smallest integer $k$ such that $G$ has an adjacent vertex distinguishing edge $k$-coloring. In this paper, we show that every bipartite planar graph $G$ with maximum degree $\Delta\ge 7$ and without isolated edges has $\chi_{a

Walk-Powers and Homomorphism Bounds of Planar Signed Graphs

As an extension of the Four-Color Theorem it is conjectured by the first author that every planar graph of odd-girth at least $$2k+1$$2k+1 admits a homomorphism to the projective cube of dimension 2k, i.e., the Cayley graph $${\mathcal {PC}}(2k)=({\mathbb {Z}}_2^{2k}, \{e_1, e_2,$$PC(2k)=(Z22k,{e1,e2,$$\ldots ,e_{2k}, J\})$$�,e2k,J}) where the $$e_i$$ei's are the standard basis vectors of $${\mathbb {Z}}_2^d$$Z2d and J is the all 1 vector. Noting that $${\mathcal {PC}}(2k)$$PC(2k) itself is of odd-girth $$2k+1$$2k+1, in this work we show that if the conjecture is true, then $${\mathcal {PC}}(2k)$$PC(2k) is an optimal such graph both with respect to the number of vertices and the number of edges. The result is obtained using the notion of walk-power of graphs and their clique numbers. An analogous result is proved for signed bipartite planar graphs of unbalanced-girth 2k. The work is presented in the uniform framework of planar consistent signed graphs.

The k-hop connected dominating set problem: hardness and polyhedra

Let G=(V,E) be a connected graph, and k a positive integer. A subset D⊆V is a k-hop connected dominating set (k-CDS) if the subgraph of G induced by D is connected and, for every vertex v in G, there is a vertex u in D such that the distance between v and u is at most k. We study the problem of finding a minimum k-hop connected dominating set, denoted by the acronym Mink-CDS. Firstly, we prove that Mink-CDS is NP-hard on planar bipartite graphs of maximum degree 4 and on planar biconnected graphs of maximum degree 5. We present an inapproximability threshold for Mink-CDS on bipartite and on (1, 2)-split graphs, and we also prove that Mink-CDS is APX-hard on bipartite graphs of maximum degree 4. These results are shown to hold for every positive integer k. For k=1, the classical minimum connected dominating set problem, we present an integer linear programming formulation and show some classes of inequalities that define facets of the corresponding polytope. We also present an approximation algorithm for this case.

Bipartite rigidity

We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers k , l k,l the notions of ( k , l ) (k,l) -rigid and ( k , l ) (k,l) -stress free bipartite graphs. This theory coincides with the study of Babson–Novik’s balanced shifting restricted to graphs. We establish bipartite analogs of the cone, contraction, deletion, and gluing lemmas, and apply these results to derive a bipartite analog of the rigidity criterion for planar graphs. Our result asserts that for a planar bipartite graph G G its balanced shifting, G b G^b , does not contain K 3 , 3 K_{3,3} ; equivalently, planar bipartite graphs are generically ( 2 , 2 ) (2,2) -stress free. We also discuss potential applications of this theory to Jockusch’s cubical lower bound conjecture and to upper bound conjectures for embedded simplicial complexes.

Three ways to cover a graph

We consider the problem of covering an input graphH with graphs from a fixed covering classG. The classical covering number of H with respect to G is the minimum number of graphs from G needed to cover the edges of H without covering non-edges of H. We introduce a unifying notion of three covering parameters with respect to G, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures “how far” H is from G in a different way. Whereas the folded covering number has been investigated thoroughly for some covering classes, e.g., interval graphs and planar graphs, the local covering number has received little attention.We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs. The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity. As input graphs we consider graphs of bounded degeneracy, bounded degree, bounded tree-width or bounded simple tree-width, as well as outerplanar, planar bipartite, and planar graphs. For several pairs of an input class and a covering class we determine exactly the maximum ordinary, local, and folded covering number of an input graph with respect to that covering class.

Anti-forcing numbers of perfect matchings of graphs

We define the anti-forcing number of a perfect matching M of a graph G as the minimal number of edges of G whose deletion results in a subgraph with a unique perfect matching M, denoted by af(G,M). The anti-forcing number of a graph proposed by Vukičević and Trinajstić in Kekulé structures of molecular graphs is in fact the minimum anti-forcing number of perfect matchings. For plane bipartite graph G with a perfect matching M, we obtain a minimax result: af(G,M) equals the maximal number of M-alternating cycles of G where any two either are disjoint or intersect only at edges in M. For a hexagonal system H, we show that the maximum anti-forcing number of H equals the Fries number of H. As a consequence, we have that the Fries number of H is between the Clar number of H and twice. Further, some extremal graphs are discussed.

Dimers, webs, and positroids

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally non-negative Grassmannian. Considering pairing probabilities for the double-dimer model gives rise to Grassmann analogs of Rhoades and Skandera's Temperley–Lieb immanants. The same problem for the (probably novel) triple-dimer model gives rise to the combinatorics of Kuperberg's webs and Grassmann analogs of Pylyavskyy's web immanants. This draws a connection between the square move of plabic graphs (or urban renewal of planar bipartite graphs), and Kuperberg's square reduction of webs. Our results also suggest that canonical-like bases might be applied to the dimer model. We furthermore show that these functions on the Grassmannian are compatible with restriction to positroid varieties. Namely, our construction gives bases for the degree 2 and degree 3 components of the homogeneous coordinate ring of a positroid variety that are compatible with the cyclic group action.

Bipartite minors

We introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain K3,3 as a bipartite minor. Similarly, we provide a forbidden minor characterization for outerplanar graphs and forests. We then establish a recursive characterization of bipartite (2,2)-Laman graphs — a certain family of graphs that contains all maximal bipartite planar graphs.

Completing Partial Proper Colorings using Hall's Condition

In the context of list-coloring the vertices of a graph, Hall's condition is a generalization of Hall's Marriage Theorem and is necessary (but not sufficient) for a graph to admit a proper list-coloring. The graph $G$ with list assignment $L$ satisfies Hall's condition if for each subgraph $H$ of $G$, the inequality $|V(H)| \leq \sum_{\sigma \in \mathcal{C}} \alpha(H(\sigma, L))$ is satisfied, where $\mathcal{C}$ is the set of colors and $\alpha(H(\sigma, L))$ is the independence number of the subgraph of $H$ induced on the set of vertices having color $\sigma$ in their lists. A list assignment $L$ to a graph $G$ is called Hall if $(G,L)$ satisfies Hall's condition. A graph $G$ is Hall $m$-completable for some $m \geq \chi(G)$ if every partial proper $m$-coloring of $G$ whose corresponding list assignment is Hall can be extended to a proper coloring of $G$. In 2011, Bobga et al. posed the following questions: (1) Are there examples of graphs that are Hall $m$-completable, but not Hall $(m+1)$-completable for some $m \geq 3$? (2) If $G$ is neither complete nor an odd cycle, is $G$ Hall $\Delta(G)$-completable? This paper establishes that for every $m \geq 3$, there exists a graph that is Hall $m$-completable but not Hall $(m+1)$-completable and also that every bipartite planar graph $G$ is Hall $\Delta(G)$-completable.