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

ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS

Proceedings of the YSU A: Physical and Mathematical Sciences ◽

10.46991/pysu:a/2020.54.3.137 ◽

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.

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Reconstructing a Graph from its Neighborhood Lists

Combinatorics Probability Computing ◽

10.1017/s0963548300000535 ◽

1993 ◽

Vol 2 (2) ◽

pp. 103-113 ◽

Cited By ~ 10

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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Linear-Time Algorithm for the Paired-Domination Problem in Convex Bipartite Graphs

Theory of Computing Systems ◽

10.1007/s00224-011-9378-8 ◽

2011 ◽

Vol 50 (4) ◽

pp. 721-738 ◽

Cited By ~ 8

Author(s):

Ruo-Wei Hung

Keyword(s):

Linear Time ◽

Time Algorithm ◽

Bipartite Graphs ◽

Linear Time Algorithm ◽

Domination Problem ◽

Paired Domination ◽

Convex Bipartite Graphs

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Consecutive Colouring of Oriented Graphs

Results in Mathematics ◽

10.1007/s00025-021-01505-3 ◽

2021 ◽

Vol 76 (4) ◽

Author(s):

Marta Borowiecka-Olszewska ◽

Ewa Drgas-Burchardt ◽

Nahid Yelene Javier-Nol ◽

Rita Zuazua

Keyword(s):

Planar Graphs ◽

Maximum Degree ◽

Bipartite Graphs ◽

Complete Graphs ◽

Oriented Graphs ◽

Complete Problem ◽

Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

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