On the Maximal Shortest Paths Cover Number

• Main Authors: Iztok Peterin, Gabriel Semanišin
• Format: Article
• Language: English
• Published: MDPI AG 2021-07-01
• Series: Mathematics
• Subjects: distance; shortest path; maximal shortest paths cover; tree
• Online Access: https://www.mdpi.com/2227-7390/9/14/1592

A shortest path <i>P</i> of a graph <i>G</i> is maximal if <i>P</i> is not contained as a subpath in any other shortest path. A set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a maximal shortest paths cover if every maximal shortest path of <i>G</i> contains a vertex of <i>S</i>. The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We show that it is NP-hard to determine <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We establish a connection between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> and several other graph parameters. We present a linear time algorithm that computes exact value for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ξ</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a tree <i>T</i>.