The Kuratowski–Wagner Theorem asserts that a graph is planar if and only if it does not have either K3,3 or K5 as a minor. Using this, Wagner obtained a precise description of all graphs with no ...K3,3‐minor and all graphs with no K5‐minor. Similar results have been achieved for the class of graphs with no H‐minor for a number of small graphs H. In this paper we give a precise structure theorem for graphs which do not contain K3,3 as an immersion. This strengthens an earlier theorem of Giannopoulou, Kamiński, and Thilikos that gives a rough description of the class of graphs with no K3,3 or K5 immersion.
This paper gives a precise structure theorem for the class of graphs which do not contain
W
4 as an immersion. This strengthens a previous result of Belmonte et al. that gives a rough description of ...this class. In fact, we prove a stronger theorem concerning rooted immersions of
W
4 where one terminal is specified in advance. This stronger result is key in a forthcoming structure theorem for graphs with no
K
3
,
3 immersion.
Abstract
This paper gives a precise structure theorem for the class of graphs which do not contain as an immersion. This strengthens a previous result of Belmonte et al. that gives a rough ...description of this class. In fact, we prove a stronger theorem concerning rooted immersions of where one terminal is specified in advance. This stronger result is key in a forthcoming structure theorem for graphs with no immersion.
We establish splitter theorems for graph immersions for two families of graphs, \(k\)-edge-connected graphs, with \(k\) even, and 3-edge-connected, internally 4-edge-connected graphs. As a corollary, ...we prove that every \(3\)-edge-connected, internally \(4\)-edge-connected graph on at least seven vertices that immerses \(K_5\) also has \(K_{3,3}\) as an immersion.
This paper gives a precise structure theorem for the class of graphs which do not contain \(W_4\) as an immersion. This strengthens a previous result of Belmonte at al. that gives a rough description ...of this class. In fact, we prove a stronger theorem concerning rooted immersions of \(W_4\) where one terminal is specified in advance. This stronger result is key in a forthcoming structure theorem for graphs with no \(K_{3,3}\) immersion.
The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either \(K_{3,3}\) or \(K_5\) as a minor. Using this Wagner obtained a precise description of all graphs ...with no \(K_{3,3}\) minor and all graphs with no \(K_5\) minor. Similar results have been achieved for the class of graphs with no \(H\)-minor for a number of small graphs \(H\). In this paper we give a precise structure theorem for graphs which do not contain \(K_{3,3}\) as an immersion. This strengthens an earlier theorem of Giannopoulou, Kami\'{n}ski, and Thilikos that gives a rough description of the class of graphs with no \(K_{3,3}\) or \(K_5\) immersion.
Understanding the underlying causes of congenital anomalies (CAs) can be a complex diagnostic journey. We aimed to assess the efficiency of exome sequencing (ES) and chromosomal microarray analysis ...(CMA) in patients with CAs among a population with a high fraction of consanguineous marriage. Depending on the patient's symptoms and family history, karyotype/Quantitative Fluorescence- Polymerase Chain Reaction (QF-PCR) (n = 84), CMA (n = 81), ES (n = 79) or combined CMA and ES (n = 24) were performed on 168 probands (66 prenatal and 102 postnatal) with CAs. Twelve (14.28%) probands were diagnosed by karyotype/QF-PCR and seven (8.64%) others were diagnosed by CMA. ES findings were conclusive in 39 (49.36%) families, and 61.90% of them were novel variants. Also, 64.28% of these variants were identified in genes that follow recessive inheritance in CAs. The diagnostic rate (DR) of ES was significantly higher than that of CMA in children from consanguineous families (P = 0·0001). The highest DR by CMA was obtained in the non-consanguineous postnatal subgroup and by ES in the consanguineous prenatal subgroup. In a population that is highly consanguineous, our results suggest that ES may have a higher diagnostic yield than CMA and should be considered as the first-tier test in the evaluation of patients with congenital anomalies.
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