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Strictly chordal graph

An undirected graph G is called strictly chordal if it is chordal and any cycle of even length (⩾6 {\ displaystyle \ geqslant 6} {\ displaystyle \ geqslant 6} ) in G has an odd chord , that is, an edge that connects the two vertices of the cycle at an odd distance (> 1) from each other [1] .

Content

Description

Strictly chordal graphs are described by forbidden graphs as graphs that do not contain an generated cycle longer than three or n -suns (n⩾3 {\ displaystyle n \ geqslant 3}   ) as a generated subgraph [2] [3] [4] . The n- Sun is a chordal graph with 2 n vertices divided into two subsets.U={uone,u2,...} {\ displaystyle U = \ {u_ {1}, u_ {2}, \ dots \}}   andW={wone,w2,...} {\ displaystyle W = \ {w_ {1}, w_ {2}, \ dots \}}   so that each vertex w i of W has exactly two neighbors, u i andu(i+one)modn {\ displaystyle u _ {(i + 1) \ mod n}}   . n - the Sun can not be strictly chordal, since the cycleuonewoneu2w2 {\ displaystyle u_ {1} w_ {1} u_ {2} w_ {2}}   ... has no odd chords.

Strictly chordal graphs can be described as graphs that have a strict perfect exception order, a vertex order such that the neighbors of any vertex that follow the order form a clique , and such that for anyi<j<k<l {\ displaystyle i <j <k <l}   , if the i -th vertex is adjacent to the kth and lth vertex, and the jth and kth vertices are adjacent, then the jth and lth vertices must be adjacent [3] [5] .

A graph is strictly chordal if and only if any of its generated subgraphs has a simple vertex, that is, a vertex whose neighbors are linearly ordered in the order of inclusion [3] [6] . Also, a graph is strictly chordalen if and only if it is chordalen and any cycle of length five or more has a 2-chord triangle, that is, a triangle formed by two chords and an edge of the cycle [7] .

A graph is strictly chordal if and only if any of its generated subgraphs is a [8] .

Strictly chordal graphs can be described in terms of the number of complete subgraphs to which the edge belongs [9] . Another description is given in the article by De Caria and Mackey [10] .

Recognition

You can determine whether a graph is strictly chordal, in polynomial time, by repeatedly searching for and removing a simple vertex. If this process excludes all vertices of the graph, the graph must be strictly chordal. Otherwise, the process does not find a subgraph without a simple vertex, and in this case the initial graph cannot be strictly chordal. For a strictly chordal graph, the order in which vertices are deleted in this process is called the strict perfect order of elimination [3] .

Alternative algorithms are known that can determine whether a graph is strictly chordal and, if so, build a strict perfect exclusion order more efficiently, over time.O(min(n2,(n+m)log⁡n)) {\ displaystyle O (\ min (n ^ {2}, (n + m) \ log n))}   for a graph with n vertices and m edges [11] [12] [13] .

Subclasses

An important subclass (based on phylogeny ) is the class of k - , that is, graphs formed from leaves of a tree by connecting two leaves with an edge, if the distance in the tree does not exceed k . A leaf degree is a graph that is a k -leaf degree for some k . Since the degrees of strictly chordal graphs are strictly chordal and the trees are strictly chordal, it follows that the leaf degrees are strictly chordal. They form a proper subclass of strictly chordal graphs, which, in turn, includes as 2-leaf degrees [14] . Another important subclass of strictly chordal graphs is interval graphs . The article by Branshtedt, Hudt, Mancini, and Wagner [15] showed that interval graphs and a larger class of root oriented paths are leaf powers.

Algorithmic problems

Since strictly chordal graphs are simultaneously chordal and , various NP-complete problems, such as the independent set problem, the clique problem , coloring , the clipping floor problem , the dominant set, and the Steiner minimal tree problem can be solved effectively for strictly chordal graphs. The graph isomorphism problem is GI-complete [16] for strictly chordal graphs [17] . The search for Hamiltonian cycles for a strictly chordal split graphs remains an NP-complete problem [18] .

Notes

  1. ↑ Brandstädt, Le, Spinrad, 1999 , p. 43, Definition 3.4.1.
  2. ↑ Chang, 1982 .
  3. ↑ 1 2 3 4 Farber, 1983 .
  4. ↑ Brandstädt, Le, Spinrad, 1999 , p. 112, Theorem 7.2.1.
  5. ↑ Brandstädt, Le, Spinrad, 1999 , p. 77, Theorem 5.5.1.
  6. ↑ Brandstädt, Le, Spinrad, 1999 , p. 78, Theorem 5.5.2.
  7. ↑ Dahlhaus, Manuel, Miller, 1998 .
  8. ↑ Brandstädt, Dragan, Chepoi, Voloshin, 1998 , p. 444, Corollary 3.
  9. ↑ McKee, 1999 .
  10. ↑ De Caria, McKee, 2014 .
  11. ↑ Lubiw, 1987 .
  12. ↑ Paige, Tarjan, 1987 .
  13. ↑ Spinrad, 1993 .
  14. ↑ Nishimura, Ragde, Thilikos, 2002 .
  15. ↑ Brandstädt, Hundt, Mancini, Wagner, 2010 .
  16. ↑ The article introduces a new class of completeness - Graph Isomorphism completeness
  17. ↑ Uehara, Toda, Nagoya, 2005 .
  18. ↑ Müller, 1996 .

Literature

  • Andreas Brandstädt, Feodor Dragan, Victor Chepoi, Vitaly Voloshin. Dually Chordal Graphs // SIAM Journal on Discrete Mathematics . - 1998. - Vol . 11 . - p . 437–455 . - DOI : 10.1137 / s0895480193253415 .
  • Andreas Brandstädt, Christian Hundt, Federico Mancini, Peter Wagner. Rooted directed paths graphs are leaf powers // Discrete Mathematics . - 2010. - T. 310 . - p . 897–910 . - DOI : 10.1016 / j.disc.2009.10.006 . .
  • Andreas Brandstädt, Van Bang Le. Recognition of 3-leaf powers // Information Processing Letters . - 2006. - T. 98 . - pp . 133–138 . - DOI : 10.1016 / j.ipl.2006.01.004 . .
  • Andreas Brandstädt, Van Bang Le, Sritharan R. R. Acquisition of the 4th leaf powers // ACM Transactions on Algorithms . - 2008. - Vol . 5 . - S. Article 11 . .
  • Andreas Brandstädt, Van Bang Le, Jeremy Spinrad. Graph Classes: A Survey. - SIAM Monographs on Discrete Mathematics and Applications, 1999. - ISBN 0-89871-432-X .
  • Chang GJ K- Domination and Graph Covering Problems. - Cornell University, 1982. - (Ph.D. thesis).
  • Dahlhaus E., Manuel PD, Miller M. A characterization of strongly chordal graphs // Discrete Mathematics . - 1998. - V. 187 , no. 1–3 . - p . 269-271 . - DOI : 10.1016 / S0012-365X (97) 00268-9 .
  • De Caria P., McKee TA Maxclique and unit disk characterizations of strongly chordal graphs // Discussion Mathematicae Graph Theory. - 2014. - T. 34 . - p . 593–602 . - DOI : 10.7151 / dmgt.1757 . .
  • Farber M. Characterizations of strongly chordal graphs // Discrete Mathematics . - 1983. - T. 43 . - p . 173–189 . - DOI : 10.1016 / 0012-365X (83) 90154-1 . .
  • Anna Lubiw. Doubly lexical orderings of matrices // SIAM Journal on Computing . - 1987. - V. 16 . - p . 854–879 . - DOI : 10.1137 / 0216057 .
  • McKee TA A new characterization of strongly chordal graphs // Discrete Mathematics . - 1999. - V. 205 , no. 1–3 . - p . 245–247 . - DOI : 10.1016 / S0012-365X (99) 00107-7 .
  • Müller H. Hamiltonian Circuits in Chordal Bipartite Graphs // Discrete Mathematics . - 1996. - T. 156 . - p . 291–298 . - DOI : 10.1016 / 0012-365x (95) 00057-4 .
  • Nishimura N., Ragde P., Thilikos DM On the powers of leaf-labeled trees // Journal of Algorithms . - 2002. - V. 42 . - pp . 69–108 . - DOI : 10.1006 / jagm.2001.1195 .
  • Paige R., Tarjan RE Partition algorithms // SIAM Journal on Computing . - 1987. - V. 16 . - p . 973–989 . - DOI : 10.1137 / 0216062 .
  • Rautenbach D. Some remarks about leaf roots // Discrete Mathematics . - 2006. - T. 306 . - p . 1456–1461 . - DOI : 10.1016 / j.disc.2006.03.030 .
  • Spinrad J. Doubly lexical ordering of dense 0–1 matrices // Information Processing Letters . - 1993. - T. 45 . - p . 229–235 . - DOI : 10.1016 / 0020-0190 (93) 90209-R .
  • Uehara R., Toda S., Nagoya T. Graph isomorphism completeness for chordal bipartite and strongly chordal graphs // Discrete Applied Mathematics . - 2005. - T. 145 . - p . 479–482 . - DOI : 10.1016 / j.dam.2004.06.008 . .
Source - https://ru.wikipedia.org/w/index.php?title=Strict_Hordal_graph&oldid=99255864


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