scholarly journals A Characterization of 2-Tree Proper Interval 3-Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
David E. Brown ◽  
Breeann M. Flesch
Keyword(s):  
Nonempty Intersection ◽  
Intersection Graph ◽  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs ◽  
Different Color

An interval p-graph is the intersection graph of a collection of intervals which have been colored with p different colors with edges corresponding to nonempty intersection of intervals from different color classes. We characterize the class of 2-trees which are interval 3-graphs via a list of three graphs and three infinite families of forbidden induced subgraphs.

scholarly journals Linear Time Recognition of P4-Indifferent Graphs

1999 ◽  
Vol 6 (38) ◽  
Author(s):  
Romeo Rizzi
Keyword(s):  
Key Words ◽  
Linear Time ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Induced Subgraphs ◽  
Modular Decomposition ◽  
Forbidden Induced Subgraphs ◽  
Chordless Path

<p>A simple graph is P4-indifferent if it admits a total order < on<br />its nodes such that every chordless path with nodes a, b, c, d and edges<br />ab, bc, cd has a < b < c < d or a > b > c > d. P4-indifferent graphs generalize<br /> indifferent graphs and are perfectly orderable. Recently, Hoang,<br />Maray and Noy gave a characterization of P4-indifferent graphs in<br />terms of forbidden induced subgraphs. We clarify their proof and describe<br /> a linear time algorithm to recognize P4-indifferent graphs. When<br />the input is a P4-indifferent graph, then the algorithm computes an order < as above.</p><p>Key words: P4-indifference, linear time, recognition, modular decomposition.</p><p> </p>

scholarly journals Balancedness of subclasses of circular-arc graphs

2014 ◽  
Vol Vol. 16 no. 3 (Graph Theory) ◽  
Author(s):  
Flavia Bonomo ◽  
Guillermo Duran ◽  
Martın D. Safe ◽  
Annegret K. Wagler
Keyword(s):  
Incidence Matrix ◽  
Perfect Graphs ◽  
Induced Subgraphs ◽  
Circular Arc ◽  
Forbidden Induced Subgraphs ◽  
Odd Order ◽  
International Audience ◽  
Circular Arc Graphs ◽  
Balanced Graphs

Graph Theory International audience A graph is balanced if its clique-vertex incidence matrix contains no square submatrix of odd order with exactly two ones per row and per column. There is a characterization of balanced graphs by forbidden induced subgraphs, but no characterization by mininal forbidden induced subgraphs is known, not even for the case of circular-arc graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. In this work, we characterize when a given graph G is balanced in terms of minimal forbidden induced subgraphs, by restricting the analysis to the case where G belongs to certain classes of circular-arc graphs, including Helly circular-arc graphs, claw-free circular-arc graphs, and gem-free circular-arc graphs. In the case of gem-free circular-arc graphs, analogous characterizations are derived for two superclasses of balanced graphs: clique-perfect graphs and coordinated graphs.

scholarly journals Matrix Partitions with Finitely Many Obstructions

10.37236/976 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Tomás Feder ◽  
Pavol Hell ◽  
Wing Xie
Keyword(s):  
Symmetric Matrix ◽  
Forbidden Subgraph ◽  
Input Graph ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Extra Effort ◽  
Matrix Partition ◽  
Finite Set ◽  
Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

scholarly journals A Graphlet-Based Topological Characterization of the Resting-State Network in Healthy People

2021 ◽  
Vol 15 ◽  
Author(s):  
Paolo Finotelli ◽  
Carlo Piccardi ◽  
Edie Miglio ◽  
Paolo Dulio
Keyword(s):  
Resting State ◽  
Brain Regions ◽  
Brain Network ◽  
Induced Subgraphs ◽  
Topological Characterization ◽  
Network Information ◽  
The Subject ◽  
Euclidean Distances ◽  
The Brain

In this paper, we propose a graphlet-based topological algorithm for the investigation of the brain network at resting state (RS). To this aim, we model the brain as a graph, where (labeled) nodes correspond to specific cerebral areas and links are weighted connections determined by the intensity of the functional magnetic resonance imaging (fMRI). Then, we select a number of working graphlets, namely, connected and non-isomorphic induced subgraphs. We compute, for each labeled node, its Graphlet Degree Vector (GDV), which allows us to associate a GDV matrix to each one of the 133 subjects of the considered sample, reporting how many times each node of the atlas “touches” the independent orbits defined by the graphlet set. We focus on the 56 independent columns (i.e., non-redundant orbits) of the GDV matrices. By aggregating their count all over the 133 subjects and then by sorting each column independently, we obtain a sorted node table, whose top-level entries highlight the nodes (i.e., brain regions) most frequently touching each of the 56 independent graphlet orbits. Then, by pairwise comparing the columns of the sorted node table in the top-k entries for various values of k, we identify sets of nodes that are consistently involved with high frequency in the 56 independent graphlet orbits all over the 133 subjects. It turns out that these sets consist of labeled nodes directly belonging to the default mode network (DMN) or strongly interacting with it at the RS, indicating that graphlet analysis provides a viable tool for the topological characterization of such brain regions. We finally provide a validation of the graphlet approach by testing its power in catching network differences. To this aim, we encode in a Graphlet Correlation Matrix (GCM) the network information associated with each subject then construct a subject-to-subject Graphlet Correlation Distance (GCD) matrix based on the Euclidean distances between all possible pairs of GCM. The analysis of the clusters induced by the GCD matrix shows a clear separation of the subjects in two groups, whose relationship with the subject characteristics is investigated.

scholarly journals Some New Classes of Open Distance-Pattern Uniform Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bibin K. Jose
Keyword(s):  
Nonempty Subset ◽  
Chordal Graphs ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Induced Subgraphs ◽  
Minimum Cardinality ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Strongly Chordal Graphs

Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph G can be embedded into a self-complementary odpu-graph H, such that G and G¯ are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.

scholarly journals Forbidden induced subgraphs for star-free graphs

2011 ◽  
Vol 311 (21) ◽  
pp. 2475-2484 ◽  
Author(s):  
Jun Fujisawa ◽  
Katsuhiro Ota ◽  
Kenta Ozeki ◽  
Gabriel Sueiro
Keyword(s):  
Induced Subgraphs ◽  
Forbidden Induced Subgraphs
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