ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS

2020 ◽  
Vol 54 (3 (253)) ◽  
pp. 137-145
Author(s):  
Aram H. Gharibyan ◽  
Petros A. Petrosyan
Keyword(s):  
Bipartite Graph ◽  
Open Neighborhood ◽  
Bipartite Graphs ◽  
The Other ◽  
Open Neighbourhood ◽  
Np Complete ◽  
Convex Bipartite Graphs

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.

Tree Convex Bipartite Graphs: $\mathcal{NP}$ -Complete Domination, Hamiltonicity and Treewidth

2014 ◽  
pp. 252-263 ◽  
Author(s):  
Chaoyi Wang ◽  
Hao Chen ◽  
Zihan Lei ◽  
Ziyang Tang ◽  
Tian Liu ◽  
...  
Keyword(s):  
Bipartite Graphs ◽  
Np Complete ◽  
Convex Bipartite Graphs

scholarly journals A Note on Path Factors of $(3,4)$-Biregular Bipartite Graphs

10.37236/705 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Carl Johan Casselgren
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Edge Coloring ◽  
Bipartite Graphs ◽  
The Other ◽  
Spanning Subgraph ◽  
Interval Coloring ◽  
Proper Edge Coloring

A proper edge coloring of a graph $G$ with colors $1,2,3,\dots$ is called an interval coloring if the colors on the edges incident with any vertex are consecutive. A bipartite graph is $(3,4)$-biregular if all vertices in one part have degree $3$ and all vertices in the other part have degree $4$. Recently it was proved [J. Graph Theory 61 (2009), 88-97] that if such a graph $G$ has a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in $\{2, 4, 6, 8\}$, then $G$ has an interval coloring. It was also conjectured that every simple $(3,4)$-biregular bipartite graph has such a subgraph. We provide some evidence for this conjecture by proving that a simple $(3,4)$-biregular bipartite graph has a spanning subgraph whose components are nontrivial paths with endpoints at $3$-valent vertices and lengths not exceeding $22$.

scholarly journals The Quantifier Semigroup for Bipartite Graphs

10.37236/610 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Leonard J. Schulman
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
The Other ◽  
Monotone Mappings ◽  
Ordered Semigroup ◽  
Partially Ordered

In a bipartite graph there are two widely encountered monotone mappings from subsets of one side of the graph to subsets of the other side: one corresponds to the quantifier "there exists a neighbor in the subset" and the other to the quantifier "all neighbors are in the subset." These mappings generate a partially ordered semigroup which we characterize in terms of "run-unimodal" words.

scholarly journals Detection number of bipartite graphs and cubic graphs

2014 ◽  
Vol Vol. 16 no. 3 ◽  
Author(s):  
Frederic Havet ◽  
Nagarajan Paramaguru ◽  
Rathinaswamy Sampathkumar
Keyword(s):  
Positive Integer ◽  
Bipartite Graph ◽  
Connected Graph ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
International Audience ◽  
Np Complete

International audience For a connected graph G of order |V(G)| ≥3 and a k-labelling c : E(G) →{1,2,…,k} of the edges of G, the code of a vertex v of G is the ordered k-tuple (ℓ1,ℓ2,…,ℓk), where ℓi is the number of edges incident with v that are labelled i. The k-labelling c is detectable if every two adjacent vertices of G have distinct codes. The minimum positive integer k for which G has a detectable k-labelling is the detection number det(G) of G. In this paper, we show that it is NP-complete to decide if the detection number of a cubic graph is 2. We also show that the detection number of every bipartite graph of minimum degree at least 3 is at most 2. Finally, we give some sufficient condition for a cubic graph to have detection number 3.

Two-dimensional partial orderings: Undecidability

1980 ◽  
Vol 45 (1) ◽  
pp. 133-143 ◽  
Author(s):  
Alfred B. Manaster ◽  
Joseph G. Rosenstein
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
The Other ◽  
Two Dimensional ◽  
One Dimensional ◽  
Recursive Model ◽  
Linear Orderings ◽  
Other Hand ◽  
Partial Orderings ◽  
Planar Lattices

In this paper we examine the class of two-dimensional partial orderings from the perspective of undecidability. We shall see that from this perspective the class of 2dpo's is more similar to the class of all partial orderings than to its one-dimensional subclass, the class of all linear orderings. More specifically, we shall describe an argument which lends itself to proofs of the following four results:(A) the theory of 2dpo's is undecidable:(B) the theory of 2dpo's is recursively inseparable from the set of sentences refutable in some finite 2dpo;(C) there is a sentence which is true in some 2dpo but which has no recursive model;(D) the theory of planar lattices is undecidable.It is known that the theory of linear orderings is decidable (Lauchli and Leonard [4]). On the other hand, the theories of partial orderings and lattices were shown to be undecidable by Tarski [14], and that each of these theories is recursively inseparable from its finitely refutable statements was shown by Taitslin [13]. Thus, the complexity of the theories of partial orderings and lattices is, by (A), (B) and (D), already reflected in the 2dpo's and planar lattices.As pointed out by J. Schmerl, bipartite graphs can be coded into 2dpo's, so that (A) and (B) could also be obtained by applying a Rabin-Scott style argument [9] to Rogers' result [11] that the theory of bipartite graphs is undecidable and to Lavrov's result [5] that the theory of bipartite graphs is recursively inseparable from the set of sentences refutable in some finite bipartite graph. (However, (C) and (D) do not seem to follow from this type of argument.)

scholarly journals Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Algorithmica ◽  
2022 ◽  
Author(s):  
Boris Klemz ◽  
Günter Rote
Keyword(s):  
Bipartite Graph ◽  
Linear Time ◽  
Bipartite Graphs ◽  
Compact Representation ◽  
Maximum Cardinality ◽  
Induced Matching ◽  
Running Time ◽  
Induced Matchings ◽  
A Chain ◽  
Convex Bipartite Graphs

AbstractA bipartite graph $$G=(U,V,E)$$ G = ( U , V , E ) is convex if the vertices in V can be linearly ordered such that for each vertex $$u\in U$$ u ∈ U , the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in $$O(n+m)$$ O ( n + m ) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex $$u\in U$$ u ∈ U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in $$O(n+m)$$ O ( n + m ) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of $$O(n^2)$$ O ( n 2 ) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. 10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.

scholarly journals Transversals and Bipancyclicity in Bipartite Graph Families

10.37236/9489 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Peter Bradshaw
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
The Other ◽  
Color Class ◽  
Degree Conditions ◽  
Graph Families ◽  
Balanced Bipartition

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geqslant 4$, every balanced bipartite graph on $2n$ vertices in which each vertex in one color class has degree greater than $\frac{n}{2}$ and each vertex in the other color class has degree at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this theorem in the setting of graph transversals. Namely, we show that given a family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$ vertices with a common balanced bipartition, if each graph of $\mathcal G$ has minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree at least $\frac{n}{2}$ in the other color class, then there exists a cycle on $X$ of each even length $4 \leqslant \ell \leqslant 2n$ that uses at most one edge from each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of $n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same degree conditions, there exists a perfect matching on $X$ that uses exactly one edge from each graph of $\mathcal G$.

scholarly journals Partitioning to three matchings of given size is NP-complete for bipartite graphs

2014 ◽  
Vol 6 (2) ◽  
pp. 206-209 ◽  
Author(s):  
Dömötör Pálvölgyi
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Perfect Matching ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Np Complete

Abstract We show that the problem of deciding whether the edge set of a bipartite graph can be partitioned into three matchings, of size k1, k2 and k3 is NP-complete, even if one of the matchings is required to be perfect. We also show that the problem of deciding whether the edge set of a simple graph contains a perfect matching and a disjoint matching of size k or not is NP-complete, already for bipartite graphs with maximum degree 3. It also follows from our construction that it is NP-complete to decide whether in a bipartite graph there is a perfect matching and a disjoint matching that covers all vertices whose degree is at least 2.

scholarly journals A Note on $\mathtt{V}$-free 2-matchings

10.37236/5258 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Kristóf Bérczi ◽  
Attila Bernáth ◽  
Máté Vizer
Keyword(s):  
Bipartite Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Hypergraph Matching ◽  
Path Packing ◽  
Np Complete

Motivated by a conjecture of Liang, we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of the hypergraph matching problem. We close the paper by showing that it is NP-complete to decide whether one of the color classes of a bipartite graph can be covered by a $\mathtt{V}$-free $2$-matching.

Reconstructing a Graph from its Neighborhood Lists

1993 ◽  
Vol 2 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Martin Aigner ◽  
Eberhard Triesch
Keyword(s):  
Decision Problem ◽  
Graph Isomorphism ◽  
Bipartite Graphs ◽  
Labeled Graph ◽  
Np Complete

Associate to a finite labeled graph G(V, E) its multiset of neighborhoods (G) = {N(υ): υ ∈ V}. We discuss the question of when a list is realizable by a graph, and to what extent G is determined by (G). The main results are: the decision problem is NP-complete; for bipartite graphs the decision problem is polynomially equivalent to Graph Isomorphism; forests G are determined up to isomorphism by (G); and if G is connected bipartite and (H) = (G), then H is completely described.

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