scholarly journals Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case

10.37236/613 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Teresa Sousa
Keyword(s):  
Bipartite Graph ◽  
Minimum Weight ◽  
Total Weight ◽  
Combinatorial Theory ◽  
Single Edge ◽  
Asymptotic Value ◽  
Large N ◽  
Bipartite Case

Given graphs $G$ and $H$ and a positive number $b$, a weighted $(H,b)$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms an $H$-subgraph. We assign a weight of $b$ to each $H$-subgraph in the decomposition and a weight of 1 to single edges. The total weight of the decomposition is the sum of the weights of all elements in the decomposition. Let $\phi(n,H,b)$ be the the smallest number such that any graph $G$ of order $n$ admits an $(H,b)$-decomposition with weight at most $\phi(n,H,b)$. The value of the function $\phi(n,H,b)$ when $b=1$ was determined, for large $n$, by Pikhurko and Sousa [Minimum $H$-Decompositions of Graphs, Journal of Combinatorial Theory, B, 97 (2007), 1041–1055.] Here we determine the asymptotic value of $\phi(n,H,b)$ for any fixed bipartite graph $H$ and any value of $b$ as $n$ tends to infinity.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Maximal Flat Antichains of Minimum Weight

10.37236/158 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Martin Grüttmüller ◽  
Sven Hartmann ◽  
Thomas Kalinowski ◽  
Uwe Leck ◽  
Ian T. Roberts
Keyword(s):  
Minimum Weight ◽  
Total Weight ◽  
Minimum Size

We study maximal families ${\cal A}$ of subsets of $[n]=\{1,2,\dots,n\}$ such that ${\cal A}$ contains only pairs and triples and $A\not\subseteq B$ for all $\{A,B\}\subseteq{\cal A}$, i.e. ${\cal A}$ is an antichain. For any $n$, all such families ${\cal A}$ of minimum size are determined. This is equivalent to finding all graphs $G=(V,E)$ with $|V|=n$ and with the property that every edge is contained in some triangle and such that $|E|-|T|$ is maximum, where $T$ denotes the set of triangles in $G$. The largest possible value of $|E|-|T|$ turns out to be equal to $\lfloor(n+1)^2/8\rfloor$. Furthermore, if all pairs and triples have weights $w_2$ and $w_3$, respectively, the problem of minimizing the total weight $w({\cal A})$ of ${\cal A}$ is considered. We show that $\min w({\cal A})=(2w_2+w_3)n^2/8+o(n^2)$ for $3/n\leq w_3/w_2=:\lambda=\lambda(n) < 2$. For $\lambda\ge 2$ our problem is equivalent to the (6,3)-problem of Ruzsa and Szemerédi, and by a result of theirs it follows that $\min w({\cal A})=w_2n^2/2+o(n^2)$.

scholarly journals Embedding Spanning Bipartite Graphs of Small Bandwidth

2012 ◽  
Vol 22 (1) ◽  
pp. 71-96 ◽  
Author(s):  
FIACHRA KNOX ◽  
ANDREW TREGLOWN
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Type Result ◽  
Bounded Degree ◽  
Expansion Property ◽  
Small Bandwidth ◽  
Bipartite Case

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

scholarly journals Secure Italian domination in graphs

2020 ◽  
Author(s):  
M. Dettlaff ◽  
M. Lemańska ◽  
J. A. Rodríguez-Velázquez
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Total Weight ◽  
Np Hard ◽  
Domination In Graphs ◽  
Dominating Function

Abstract An Italian dominating function (IDF) on a graph G is a function $$f:V(G)\rightarrow \{0,1,2\}$$ f : V ( G ) → { 0 , 1 , 2 } such that for every vertex v with $$f(v)=0$$ f ( v ) = 0 , the total weight of f assigned to the neighbours of v is at least two, i.e., $$\sum _{u\in N_G(v)}f(u)\ge 2$$ ∑ u ∈ N G ( v ) f ( u ) ≥ 2 . For any function $$f:V(G)\rightarrow \{0,1,2\}$$ f : V ( G ) → { 0 , 1 , 2 } and any pair of adjacent vertices with $$f(v) = 0$$ f ( v ) = 0 and u with $$f(u) > 0$$ f ( u ) > 0 , the function $$f_{u\rightarrow v}$$ f u → v is defined by $$f_{u\rightarrow v}(v)=1$$ f u → v ( v ) = 1 , $$f_{u\rightarrow v}(u)=f(u)-1$$ f u → v ( u ) = f ( u ) - 1 and $$f_{u\rightarrow v}(x)=f(x)$$ f u → v ( x ) = f ( x ) whenever $$x\in V(G){\setminus }\{u,v\}$$ x ∈ V ( G ) \ { u , v } . A secure Italian dominating function on a graph G is defined as an IDF f which satisfies that for every vertex v with $$f(v)=0$$ f ( v ) = 0 , there exists a neighbour u with $$f(u)>0$$ f ( u ) > 0 such that $$f_{u\rightarrow v}$$ f u → v is an IDF. The weight of f is $$\omega (f)=\sum _{v\in V(G) }f(v)$$ ω ( f ) = ∑ v ∈ V ( G ) f ( v ) . The minimum weight among all secure Italian dominating functions on G is the secure Italian domination number of G. This paper is devoted to initiating the study of the secure Italian domination number of a graph. In particular, we prove that the problem of finding this parameter is NP-hard and we obtain general bounds on it. Moreover, for certain classes of graphs, we obtain closed formulas for this novel parameter.

Signed domination and Mycielski’s structure in graphs

2020 ◽  
Vol 54 (4) ◽  
pp. 1077-1086
Author(s):  
Arezoo N. Ghameshlou ◽  
Athena Shaminezhad ◽  
Ebrahim Vatandoost ◽  
Abdollah Khodkar
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Signed Domination ◽  
Signed Domination Number ◽  
Dominating Function ◽  
Domination Numbers

Let G = (V, E) be a graph. The function f : V(G) → {−1, 1} is a signed dominating function if for every vertex v ∈ V(G), ∑x∈NG[v] f(x)≥1. The value of ω(f) = ∑x∈V(G) f(x) is called the weight of f. The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete bipartite graph.

Constant-approximation for minimum weight partial sensor cover

2020 ◽  
pp. 2150047
Author(s):  
Siwen Liu ◽  
Hongmin W. Du
Keyword(s):  
Time Constant ◽  
Polynomial Time ◽  
Wireless Sensors ◽  
Minimum Weight ◽  
Total Weight ◽  
Sensor Cover ◽  
Nonnegative Weight ◽  
Cover Problem

Consider a set of homogeneous wireless sensors, [Formula: see text] with nonnegative weight [Formula: see text] for each sensor [Formula: see text]. Let [Formula: see text] be a set of target points. Given a integer [Formula: see text], we study the minimum weight partial sensor cover problem, that is, find the minimum total weight subset of sensors covering at least [Formula: see text] target points in [Formula: see text]. In this paper, we show the existence of polynomial-time constant-approximation for this problem.

scholarly journals Kempe-Locking Configurations

2018 ◽  
Author(s):  
James A. Tilley
Keyword(s):  
Bipartite Graph ◽  
Systematic Search ◽  
Single Edge ◽  
Heuristic Argument ◽  
Bottom Boundary ◽  
Single Vertex ◽  
Planar Triangulation ◽  
Isomorphism Classes ◽  
Connectivity Property ◽  
Proper Subgraph

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

scholarly journals A biometric for shark dorsal fins based on boundary descriptor matching

10.29007/bd51 ◽  
2019 ◽  
Author(s):  
Taina Coleman ◽  
Jucheol Moon
Keyword(s):  
Coordinate System ◽  
Bipartite Graph ◽  
Recent Progress ◽  
Perfect Matching ◽  
Minimum Weight ◽  
Cutting Edge ◽  
Experimental Results ◽  
Visual Data ◽  
Noninvasive Methods ◽  
Animal Biometrics

Recent progress in animal biometrics has revolutionized wildlife research. Cutting edge techniques allow researchers to track individuals through noninvasive methods of recognition that are not only more reliable, but also applicable to large, hard-to-find, and otherwise difficult to observe animals. In this research, we propose a metric for boundary descriptors based on bipartite perfect matching applied in shark dorsal fins. In order to identify a shark, we first take a fin contour and transform it to a normalized coordinate system so that we can analyze images of sharks regardless of orientation and scale. Finally, we propose a metric scheme that performs a minimum weight perfect matching in a bipartite graph. The experimental results show that our metric is applicable to identify and track individuals from visual data.

scholarly journals Reducing Invertible Embedding Distortion Using Graph Matching Model

2020 ◽  
Vol 2020 (4) ◽  
pp. 21-1-21-10
Author(s):  
Hanzhou Wu ◽  
Xinpeng Zhang
Keyword(s):  
Bipartite Graph ◽  
Graph Matching ◽  
Minimum Weight ◽  
Rate Distortion ◽  
Maximum Matching ◽  
Rate Distortion Optimization ◽  
Matching Model ◽  
Matching Problem ◽  
Weighted Bipartite Graph ◽  
Embedding Distortion

Invertible embedding allows the original cover and embedded data to be perfectly reconstructed. Conventional methods use a well-designed predictor and fully exploit the carrier characteristics. Due to the diversity, it is actually hard to accurately model arbitrary covers, which limits the practical use of methods relying heavily on content characteristics. It has motivated us to revisit invertible embedding operations and propose a general graph matching model to generalize them and further reduce the embedding distortion. In the model, the rate-distortion optimization task of invertible embedding is derived as a weighted bipartite graph matching problem. In the bipartite graph, the nodes represent the values of cover elements, and the edges indicate the candidate modifications. Each edge is associated with a weight indicating the corresponding embedding distortion for the connected nodes. By solving the minimum weight maximum matching problem, we can find the optimal embedding strategy under the constraint. Since the proposed work is a general model, it can be incorporated into existing works to improve their performance, or used for designing new invertible embedding systems. We incorporate the proposed work into a part of state-of-the-arts, and experiments show that it significantly improves the rate-distortion performance. To the best knowledge of the authors, it is probably the first work studying rate-distortion optimization of invertible embedding from the perspective of graph matching model.

Further Studies on the Mechanisms for the Antithrombotic Effects of Sulfated Polysaccharides in Rabbits

1988 ◽  
Vol 60 (02) ◽  
pp. 188-192 ◽  
Author(s):  
F A Ofosu ◽  
F Fernandez ◽  
N Anvari ◽  
C Caranobe ◽  
F Dol ◽  
...  
Keyword(s):  
Antithrombin Iii ◽  
Ex Vivo ◽  
Thrombus Formation ◽  
Minimum Weight ◽  
Sulfated Polysaccharides ◽  
Pentosan Polysulfate ◽  
Heparin Cofactor Ii ◽  
Thrombin Inhibition ◽  
The Relationship ◽  
Prothrombin Activation

SummaryA recent study (Fernandez et al., Thromb. Haemostas. 1987; 57: 286-93) demonstrated that when rabbits were injected with the minimum weight of a variety of glycosaminoglycans required to inhibit tissue factor-induced thrombus formation by —80%, exogenous thrombin was inactivated —twice as fast in the post-treatment plasmas as the pre-treatment plasmas. In this study, we investigated the relationship between inhibition of thrombus formation and the extent of thrombin inhibition ex vivo. We also investigated the relationship between inhibition of thrombus formation and inhibition of prothrombin activation ex vivo. Four sulfated polysaccharides (SPS) which influence coagulation in a variety of ways were used in this study. Unfractionated heparin and the fraction of heparin with high affinity to antithrombin III potentiate the antiproteinase activity of antithrombin III. Pentosan polysulfate potentiates the activity of heparin cofactor II. At less than 10 pg/ml of plasma, all three SPS also inhibit intrinsic prothrombin activation. The fourth agent, dermatan sulfate, potentiates the activity of heparin cofactor II but fails to inhibit intrinsic prothrombin activation even at concentrations which exceed 60 pg/ml of plasma. Inhibition of thrombus formation by each sulfated polysaccharides was linearly related to the extent of thrombin inhibition achieved ex vivo. These observations confirm the utility of catalysis of thrombin inhibition as an index for assessing antithrombotic potential of glycosaminoglycans and other sulfated polysaccharides in rabbits. With the exception of pentosan polysulfate, there was no clear relationship between inhibition of thrombus formation and inhibition of prothrombin activation ex vivo.

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