An Even Simpler Linear-Time Algorithm for Verifying Minimum Spanning Trees

In this paper, we show a polynomial-time algorithm for testing [Formula: see text]-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face. Our result is based on a reduction to the planar set of spanning trees in topological multigraphs (pssttm) problem, which is defined as follows. Given a (non-planar) topological multigraph [Formula: see text] with [Formula: see text] connected components [Formula: see text], do spanning trees of [Formula: see text] exist such that no two edges in any two spanning trees cross? Kratochvíl et al. [SIAM Journal on Discrete Mathematics, 4(2): 223–244, 1991] proved that the problem is NP-hard even if [Formula: see text]; on the other hand, Di Battista and Frati presented a linear-time algorithm to solve the pssttm problem for the case in which [Formula: see text] is a [Formula: see text]-planar topological multigraph [Journal of Graph Algorithms and Applications, 13(3): 349–378, 2009]. For any embedded flat clustered graph [Formula: see text], an instance [Formula: see text] of the pssttm problem can be constructed in polynomial time such that [Formula: see text] is [Formula: see text]-planar if and only if [Formula: see text] admits a solution. We show that, if [Formula: see text] has at most two vertices per cluster on each face, then it can be tested in polynomial time whether the corresponding instance [Formula: see text] of the pssttm problem is positive or negative. Our strategy for solving the pssttm problem on [Formula: see text] is to repeatedly perform a sequence of tests, which might let us conclude that [Formula: see text] is a negative instance, and simplifications, which might let us simplify [Formula: see text] by removing or contracting some edges. Most of these tests and simplifications are performed “locally”, by looking at the crossings involving a single edge or face of a connected component [Formula: see text] of [Formula: see text]; however, some tests and simplifications have to consider certain global structures in [Formula: see text], which we call [Formula: see text]-donuts. If no test concludes that [Formula: see text] is a negative instance of the pssttm problem, then the simplifications eventually transform [Formula: see text] into an equivalent [Formula: see text]-planar topological multigraph on which we can apply the cited linear-time algorithm by Di Battista and Frati.

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A LINEAR-TIME ALGORITHM TO FIND FOUR INDEPENDENT SPANNING TREES IN FOUR CONNECTED PLANAR GRAPHS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054199000149 ◽

1999 ◽

Vol 10 (02) ◽

pp. 195-210 ◽

Cited By ~ 17

Author(s):

KAZUYUKI MIURA ◽

DAISHIRO TAKAHASHI ◽

SHIN-ICHI NAKANO ◽

TAKAO NISHIZEKI

Keyword(s):

Natural Number ◽

Planar Graph ◽

Planar Graphs ◽

Spanning Trees ◽

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Independent Spanning Trees ◽

Connected Planar Graph

Given a graph G, a designated vertex r and a natural number k, we wish to find k "independent" spanning trees of G rooted at r, that is, k spanning trees such that the k paths connecting r and any vertex v in the k trees are internally disjoint. In this paper we give a linear-time algorithm to find four independent spanning trees in a 4-connected planar graph.

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A Linear-Time Algorithm for Finding Locally Connected Spanning Trees on Circular-Arc Graphs

Algorithmica ◽

10.1007/s00453-012-9641-7 ◽

2012 ◽

Vol 66 (2) ◽

pp. 369-396 ◽

Cited By ~ 1

Author(s):

Ching-Chi Lin ◽

Gen-Huey Chen ◽

Gerard J. Chang

Keyword(s):

Spanning Trees ◽

Linear Time ◽

Time Algorithm ◽

Linear Time Algorithm ◽

Circular Arc ◽

Circular Arc Graphs

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Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽

10.3390/math9030293 ◽

2021 ◽

Vol 9 (3) ◽

pp. 293

Author(s):

Xinyue Liu ◽

Huiqin Jiang ◽

Pu Wu ◽

Zehui Shao

Keyword(s):

Linear Time ◽

Minimum Weight ◽

Domination Number ◽

Time Algorithm ◽

Simple Graph ◽

Linear Time Algorithm ◽

Domination Problem ◽

Np Complete ◽

Chordal Bipartite Graphs ◽

Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

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