Common vertex of longest cycles in circular arc graphs Hehui Wu University of Illinois at Urbana-Champaign 24rd Cumberland Conference Louisville, KY 2011 Joint work with Guantao Chen Georgia State University Hehui Wu Common vertex of longest cycles in circular arc graphs
Introduction Question (Gallai 1965) All longest paths in a connected graph have a common vertex. Example (Zimfirescu 1978) NOT TRUE: exists a twelve vertex graph without a vertex cover all the longest paths. chordal graph: a graph without induced cycles of length more than three. Conjecture (Lehel ?) All longest paths in a connected chordal graph have a common vertex. Conjecture Any three longest paths in a connected graph have a common vertex. Hehui Wu Common vertex of longest cycles in circular arc graphs
Previous results Outerplanar graph: all vertices belong to the unbounded face of a planar embedding. Theorem (Axenovich) Any three longest paths in outerplanar graph have a common vertex. intersection graph G forms from a collection of sets C : G = ( V , E ) where V = C and AB ∈ E for two elements A , B ∈ C if and only if A ∩ B � = ∅ . chordal graph: the intersection graph of subtrees of a hosting tree. interval graph: the intersection graph of intervals on the real line. circular arc graph: the intersection graph of arcs on a circle. Theorem (Balister-Gy˝ ori-Lehel-Schelp 2004) All longest paths in a connected circular arc graph have a common vertex. Theorem (Balister-Gy˝ ori-Lehel-Schelp 2004) All longest cycles in a connected interval graph have a common vertex. Hehui Wu Common vertex of longest cycles in circular arc graphs
Our main results Theorem (Chen-W.) All longest cycles in a connected chordal graph have a common vertex. spider graph: A tree with at most one vertex having degree more than two. Theorem (Chen-W.) All longest paths in the intersection graph of a collection of subtrees of a spider graph have a common vertex. Hehui Wu Common vertex of longest cycles in circular arc graphs
Nondecreasing path For a vertex X in an interval graph, let L ( X ) and R ( X ) be its left endpoint and right endpoint as an interval respectively. Given two vertices A and B in an interval graph, we say A < B iff L ( A ) < L ( B ) and R ( A ) < R ( B ). A path A 1 A 2 . . . A k in an interval graph is nondecreasing if A i � < A j whenever j < i . Lemma Any path in an interval graph can be reordered into a nondecreasing path. Given a path P with length k in an interval graph, we have the following algorithm to rearrange it into a nondecreasing path I 1 I 2 . . . I k . Algorithm: Initial: 1. S 0 = V ( P ); 2. I 1 is the vertex in S 0 with minimum right endpoint; Repeat: 3. S j = S j − 1 \ I j ; 4. I j +1 is the vertex in S j intersecting I j with minimum right endpoint; Hehui Wu Common vertex of longest cycles in circular arc graphs
Circular arc graph Theorem (Chen-W.) All longest cycles in a connected chordal graph have a common vertex. Skectch of the proof. Let A be the collection of arcs. Let M be the arcs in A which does not included in any other arc in A . Observation Arcs in M can be labeled as M 1 , M 2 , . . . , M t in the clockwise order in the circle. Observation Given any longest cycle C = A 1 A 2 . . . A m and B ∈ A . 1. If ( A i ∩ A i +1 ) ∩ B � = ∅ , then B ∈ V ( C ) . 2. If A i ⊂ B for some i, then B ∈ V ( C ) . Claim For any longest cycle C, M appear in C is consecutive in the list in M : M a , M a +1 , . . . , M b − 1 , M b . Hehui Wu Common vertex of longest cycles in circular arc graphs
Suppose the conjecture is not true. Let C 1 be a longest path with fewest elements in A . C 1 ∩ M = { A . . . B } (list in clockwise order). Let C 2 be a longest path which does not contain A . C 2 ∩ M = { C . . . D } (list in clockwise order). Observation A − > C − > B − > D − > A in clockwise order. Claim There is a nondecreasing path from A to D with length | C 1 | . Rearrange C 1 to paths: Nondecreasing: J 1 = A , J 2 , . . . , J b = B , . . . , J m . (1) Nonincreasing: I 1 = B , I 2 , . . . , I a = A , . . . , I m . (2) Rearrange C 2 to paths: Nonincreasing: J ′ 1 = D , J ′ 2 , . . . , J ′ c = C , . . . , J ′ m . (3) Nondecreasing: I ′ 1 = C , I ′ 2 , . . . , I ′ d = D , . . . , I ′ m . (4) Using (1) and (4), we get a nondecreasing path A , I a − 1 , . . . , I 2 , B , J ′ k , . . . J ′ 2 , D with length ≥ m + a − d Using (2) and (3), we get a nonincreasing path D , I ′ d − 1 , . . . , I ′ 2 , C , J j , . . . J 2 , A with length ≥ m + d − a Hehui Wu Common vertex of longest cycles in circular arc graphs
Spider graph Theorem (Chen-W.) All longest paths in the intersection graph of a collection of subtrees of a spider graph have a common vertex. Skectch of the proof. The center of the spider graph is the only vertex that have degree at least three. A branch of the spider graph is a path from the center to a leaf . Let H be the set of subtrees containing the center. Suppose the theorem is not true for some graph. Claim All the longest paths have a vertex which contains the center as a vertex in the corresponding subtree. Hehui Wu Common vertex of longest cycles in circular arc graphs
Spider graph, cont... The hub subtrees for a branch is the an element in H which has the longest subpath in this branch. A segment of a path is a maximal subpath which does not have elements in H . Observation Any segment of a path appear in a single branch. Claim For any longest path, if the hub subtree for a branch does not appear in the branch, then there is no segment appear in this branch. Consider S which is the longest segment among all the segments in all the longest paths. Let T be the corresponding hub tree for this segment. Claim T appears in all the longest paths. Proof. Suppose T does not appear in one longest path P , then neither does any element in S . Let S ′ be the last segment in this longest path. Replace S ′ by T plus S , we get a new path P ′ , which is longer. Contradiction! Hehui Wu Common vertex of longest cycles in circular arc graphs
THANK YOU! Hehui Wu Common vertex of longest cycles in circular arc graphs
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