scholarly journals Packing triangles in low degree graphs and indifference graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gordana Manić ◽  
Yoshiko Wakabayashi
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Set Packing ◽  
The Best Approximation ◽  
Low Degree ◽  
International Audience ◽  
Approximation Guarantee ◽  
Vertex Disjoint

International audience We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs.

scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

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

Mathematics ◽  
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.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

LINEAR-TIME ALGORITHMS FOR DISJOINT TWO-FACE PATHS PROBLEMS IN PLANAR GRAPHS

1996 ◽  
Vol 07 (02) ◽  
pp. 95-110 ◽  
Author(s):  
HEIKE RIPPHAUSEN-LIPA ◽  
DOROTHEA WAGNER ◽  
KARSTEN WEIHE
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Disjoint Paths ◽  
Linear Time Algorithm ◽  
Vertex Disjoint ◽  
Linear Time Algorithms

In this paper we present a linear-time algorithm for the vertex-disjoint Two-Face Paths Problem in planar graphs, i.e., the problem of finding k vertex-disjoint paths between pairs of terminals which lie on two face boundaries. The algorithm is based on the idea of finding rightmost paths with a certain property in planar graphs. Using this method, a linear-time algorithm for finding vertex-disjoint paths of a prescribed homotopy is derived. Moreover, the algorithm is modified to solve the more general linkage problem in linear time, as well.

scholarly journals Packing Three-Vertex Paths in a Subcubic Graph

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Adrian Kosowski ◽  
Michal Malafiejski ◽  
Pawel Zyliński
Keyword(s):  
Linear Time ◽  
Packing Problem ◽  
Time Algorithm ◽  
Vertex Degree ◽  
Linear Time Algorithm ◽  
Cubic Graph ◽  
Subcubic Graph ◽  
International Audience ◽  
Subcubic Graphs

International audience In our paper we consider the $P_3$-packing problem in subcubic graphs of different connectivity, improving earlier results of Kelmans and Mubayi. We show that there exists a $P_3$-packing of at least $\lceil 3n/4\rceil$ vertices in any connected subcubic graph of order $n>5$ and minimum vertex degree $\delta \geq 2$, and that this bound is tight. The proof is constructive and implied by a linear-time algorithm. We use this result to show that any $2$-connected cubic graph of order $n>8$ has a $P_3$-packing of at least $\lceil 7n/9 \rceil$ vertices.

scholarly journals Error locating: degree constraints

2021 ◽  
Author(s):  
Christopher Dennis
Keyword(s):  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Mathematical Tool ◽  
Linear Time Algorithm ◽  
Total Size ◽  
Special Cases ◽  
Locating Array ◽  
General Error

Error graphs are a useful mathematical tool for representing failing interactions in a system. This representation is used as the basis for constructing an error locating array (ELA). However, if too many errors are present in a given error graph, it may not be possible to locate all interactions. We say that a graph is locatable if an ELA can be built. Bounds on the total size of an error graph are known, bounds on the degree an error graph can have have not been considered. In this thesis we explore the maximum degree an error graph may have while still guaranteeing its locatability. We consider special cases for 3 and 4 partite error graphs as well as developing bounds on the degree of a general error graph. We describe a linear time algorithm which can be used to generate tests which have at most one failing interaction.

scholarly journals The Cycles of the Multiway Perfect Shuffle Permutation

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
John Ellis ◽  
Hongbing Fan ◽  
Jeffrey Shallit
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Equal Size ◽  
Linear Time Algorithm ◽  
Theoretic Analysis ◽  
Cycle Structure ◽  
International Audience ◽  
Perfect Shuffle

International audience The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρ _k,n: i → ki \bmod (kn+1), for 1 ≤ i ≤ kn. We uncover the cycle structure of the (k,n)-perfect shuffle permutation by a group-theoretic analysis and show how to compute representative elements from its cycles by an algorithm using O(kn) time and O((\log kn)^2) space. Consequently it is possible to realise the (k,n)-perfect shuffle via an in-place, linear-time algorithm. Algorithms that accomplish this for the 2-way shuffle have already been demonstrated.

scholarly journals Isomorphism of graph classes related to the circular-ones property

2013 ◽  
Vol Vol. 15 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Andrew R. Curtis ◽  
Min Chih Lin ◽  
Ross M. Mcconnell ◽  
Yahav Nussbaum ◽  
Francisco Juan Soulignac ◽  
...  
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Circular Arc ◽  
Graph Classes ◽  
Discrete Algorithms ◽  
International Audience ◽  
Related Graph ◽  
Circular Arc Graphs ◽  
Proper Circular Arc Graphs

Discrete Algorithms International audience We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. Our algorithm is similar to the isomorphism algorithm for interval graphs of Lueker and Booth, but works on PC trees, which are unrooted and have a cyclic nature, rather than with PQ trees, which are rooted. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ circular-arc graphs, proper circular-arc graphs and convex-round graphs.

scholarly journals A linear time algorithm for finding an Euler walk in a strongly connected 3-uniform hypergraph

2012 ◽  
Vol Vol. 14 no. 1 (Discrete Algorithms) ◽  
Author(s):  
Zbigniew Lonc ◽  
Pawel Naroski
Keyword(s):  
Linear Time ◽  
Natural Extension ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Uniform Hypergraph ◽  
Graph Theoretic ◽  
Uniform Hypergraphs ◽  
Discrete Algorithms ◽  
Strongly Connected ◽  
International Audience

Discrete Algorithms International audience By an Euler walk in a 3-uniform hypergraph H we mean an alternating sequence v(0), epsilon(1), v(1), epsilon(2), v(2), ... , v(m-1), epsilon(m), v(m) of vertices and edges in H such that each edge of H appears in this sequence exactly once and v(i-1); v(i) is an element of epsilon(i), v(i-1) not equal v(i), for every i = 1, 2, ... , m. This concept is a natural extension of the graph theoretic notion of an Euler walk to the case of 3-uniform hypergraphs. We say that a 3-uniform hypergraph H is strongly connected if it has no isolated vertices and for each two edges e and f in H there is a sequence of edges starting with e and ending with f such that each two consecutive edges in this sequence have two vertices in common. In this paper we give an algorithm that constructs an Euler walk in a strongly connected 3-uniform hypergraph (it is known that such a walk in such a hypergraph always exists). The algorithm runs in time O(m), where m is the number of edges in the input hypergraph.

scholarly journals Genus distributions of cubic series-parallel graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Jonathan L. Gross ◽  
Michal Kotrbčík ◽  
Timothy Sun
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Time Algorithm ◽  
Genus Distribution ◽  
Biconnected Components ◽  
International Audience ◽  
Parallel Graph ◽  
Cubic Series

Graph Theory International audience We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.

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