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Competition Graphs of Hamiltonian Digraphs

David R. Guichard
<span title="">1998</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
Fraughnaugh et al. proved that a graph G is the competition graph of a hamiltonian digraph possibly having loops if and only if G has an edge clique cover C = {C 1 , . . . , Cn} that has a system of distinct  ...  [2] have given characterizations for competition graphs of strongly connected digraphs and hamiltonian digraphs.  ...  Here we improve their characterizations of competition graphs of hamiltonian digraphs. (See the same paper for references to other characterizations.) 2. Hamiltonian digraphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/s089548019629735x">doi:10.1137/s089548019629735x</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ku62v23eznhpnjwo3rpmk6acqy">fatcat:ku62v23eznhpnjwo3rpmk6acqy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190303094922/http://pdfs.semanticscholar.org/c6f5/620d406266b42b5603b8b97e3fc0a6d4efd5.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c6/f5/c6f5620d406266b42b5603b8b97e3fc0a6d4efd5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/s089548019629735x"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Competition Graphs of Strongly Connected and Hamiltonian Digraphs

Kathryn F. Fraughnaugh, J. Richard Lundgren, Sarah K. Merz, John S. Maybee, Norman J. Pullman
<span title="">1995</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
graphs of loopless Hamiltonian digraphs.  ...  Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks.  ...  If T has a subtree which is the p-competition graph of a loopless Hamiltonian digraph then T is the p-competition graph of a loopless Hamiltonian digraph. Proof.  ... 
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Competition hypergraphs of digraphs with certain properties II. Hamiltonicity

Martin Sonntag, Hanns-Martin Teichert
<span title="">2008</span> <i title="Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g64zvtslcjdqncvrfbmvf422qi" style="color: black;">Discussiones Mathematicae Graph Theory</a> </i> &nbsp;
If D = (V, A) is a digraph, its competition hypergraph CH(D) has vertex set V and e ⊆ V is an edge of CH(D) iff |e| ≥ 2 and there is We give characterizations of CH(D) in case of hamiltonian digraphs D  ...  The results are closely related to the corresponding investigations for competition graphs in Fraughnaugh et al. [4] and Guichard [6] .  ...  Acknowledgement The authors would like to thank one the referees for his valuable suggestions, especially to the proof of Theorem 6 and to Lemma 7.  ... 
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Characterizing paths as m-step competition graphs

Jaromy Kuhl, Brandon Christopher Swan
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In 2005, Helleloid proved that P n is an (n − 1)-and (n − 2)-step competition graph for all n and proved further that of all connected triangle-free graphs on n vertices, only the star is an m-step competition  ...  In this paper we show that if m divides n − 1 or n − 2, then P n is an m-step competition graph and that if n ≥ 6 and n 2 ≤ m ≤ n − 3, then P n is not an m-step competition graph.  ...  [1] introduced the m-step competition graph, a generalization of the competition graph. Let D = (V , A) be a digraph and let x ∈ V .  ... 
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Page 6764 of Mathematical Reviews Vol. , Issue 98K [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We call a graph a CCE-graph if it is the competition-common enemy graph of some digraph.  ...  A digraph D is anticonnected if D contains a u-v antipath for each pair u,v of vertices of D. A digraph is anti-Hamiltonian if it contains a Hamiltonian anticycle.  ... 
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Page 4782 of Mathematical Reviews Vol. , Issue 98H [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks.  ...  Furthermore, we establish some large classes of graphs, including trees, as the p-competition graph of a loopless Hamiltonian digraph and show that interval graphs on n > 4 vertices are the 2-competition  ... 
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On the phylogeny graphs of degree-bounded digraphs [article]

Seung Chul Lee, Jihoon Choi, Suh-Ryung Kim, Yoshio Sano
<span title="2016-11-01">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Roberts: (i,j) competition graphs, Discrete Applied Mathematics, 32, (1991) 241-262] characterized acyclic digraphs each vertex of which has inderee and outdegree at most two and whose competition graphs  ...  They called acyclic digraphs each vertex of which has inderee and outdegree at most two (2,2) digraphs. In this paper, we study the phylogeny graphs of (2,2) digraphs.  ...  structure of digraphs and their corresponding competition graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1611.00246v1">arXiv:1611.00246v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/q6x5rnmsvngshcajjrj4uhffce">fatcat:q6x5rnmsvngshcajjrj4uhffce</a> </span>
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Page 653 of Mathematical Reviews Vol. , Issue 96b [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We characterize these graphs as well as establish several large classes of graphs, including chordal, interval, and some triangle-free graphs, which are competition graphs of loopless Hamiltonian digraphs  ...  Therefore it is of interest to determine which graphs are the competition graphs of strongly connected digraphs.  ... 
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Page 775 of Mathematical Reviews Vol. , Issue 99b [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Adam (H-AOS; Budapest) connectivity. 99b:05072 05C20 Guichard, David R. (1-WHIT-CS; Walla Walla, WA) Competition graphs of Hamiltonian digraphs. (English summary) SIAM J.  ...  Fraughnaugh et al. proved that a graph G is the competition graph of a Hamiltonian digraph possibly having loops if and only if G has an edge clique cover that has a system of distinct representatives  ... 
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Page 7700 of Mathematical Reviews Vol. , Issue 2001K [page]

<span title="">2001</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
A graph is k-ordered Hamiltonian if for every set S of k vertices there exists a Hamiltonian cycle which encounters the vertices of S in order.  ...  For a fixed digraph H, the problem of characterizing those digraphs G which admit a homomorphism to H has been well studied for many classes of digraphs H.  ... 
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Page 4317 of Mathematical Reviews Vol. , Issue 94h [page]

<span title="">1994</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The competition number k(G) is the smallest k such that G together with k isolated vertices is the competition graph of an acyclic digraph.  ...  Of course, these graphs can be defined for any digraph, and sev- eral new applications have expanded the study to consider various types of digraphs: for example, strongly connected or symmetric.  ... 
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Page 6614 of Mathematical Reviews Vol. , Issue 99j [page]

<span title="">1999</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
These notions are analogous to and can be considered as natural generalizations of notions of competition graph and competition number that arise from problems of ecology.  ...  Given a graph G = (V, E), the acyclic digraph D is called a phylogeny digraph for G if G is an induced subgraph of P(D) and D has no arcs from vertices outside of G to vertices in G.  ... 
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A class of acyclic digraphs with interval competition graphs

Han Hyuk Cho, Suh-Ryung Kim
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
In this paper, we show that the competition graphs of doubly partial orders are interval graphs.  ...  Let D be an acyclic digraph.  ...  They also acknowledge the anonymous referees for suggestions leading to improvements in the presentation of the results.  ... 
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Page 1396 of Mathematical Reviews Vol. , Issue 90C [page]

<span title="">1990</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In addition, it is determined which graphs are both the competition and common enemy graph of an acyclic digraph, and which are both the upper and lower bound  ...  The competition [resp. common enemy] graph of an acyclic digraph D=(V,A) is the graph G = (V,E), where xy € E if and only if x = y and for some z € V both xz and yz [resp. zx and zy] are in A.  ... 
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Page 2276 of Mathematical Reviews Vol. , Issue 85f [page]

<span title="">1985</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Roberts, Fred S. (1-RTG); Steif, Jeffrey E. (1-RTG) A characterization of competition graphs of arbitrary digraphs. Discrete Appl. Math. 6 (1983), no. 3, 323-326.  ...  Theorem 1: G is the competition graph of a digraph D (which may have loops) if and only if m(G) < |V(G)|, where m(G) is the smallest number of cliques of G which cover all edges of G.  ... 
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