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    Minimum Spanning Trees and Shortest Paths Short Notes for GATE CS

    Minimum Spanning Trees and Shortest Paths short notes for GATE CS: 9 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice q

    minimum spanning trees and shortest paths short notes

    Exam Day Summary: MST Properties

    MST Properties Cheat Sheet

    • Cut Property: Lightest edge across a cut IN the MST.
    • Cycle Property: Heaviest edge in a cycle OUT of the MST.
    • Uniqueness: All distinct weights Exactly ONE unique MST.
    • Global Min Edge: Absolute lightest edge in Always IN every MST.
    • Threshold Inclusion: Edge is in every MST weight() max weight on minimax path between its endpoints.
    • Multiple MSTs: Different trees, but identical sorted sequence of edge weights.

    Formula Matrix

    Formula Matrix

    MST Edge Count
    Exactly
    Weight Invariance
    invariant under tie-breaking
    Component Counting
    where is multigraph at weight
    Matrix Tree Theorem
    for any
    Cayley's Formula

    If You See This, Do This

    If You See This, Do This

    "lexicographical tie-breaking"
    Sort by vertex label before cycle check
    "starting vertex specified"
    Prim's; update frontier deterministically
    "number of distinct MSTs"
    Isolate components at weight ; count multigraph trees
    "forced edge in MST"
    Contract edge; run MST on remainder; add forced weight
    "all edges weight "
    Count spanning trees of (Matrix Tree / Cayley)
    "swapping edges"
    Check if they form a cycle with same max weight

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    Minimum Spanning Trees and Shortest Paths Short Notes for GATE CS

    Minimum Spanning Trees and Shortest Paths short notes for GATE CS: 9 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

    Exam Day Summary: MST Properties

    MST Properties Cheat Sheet

    • Cut Property: Lightest edge across a cut IN the MST.
    • Cycle Property: Heaviest edge in a cycle OUT of the MST.
    • Uniqueness: All distinct weights Exactly ONE unique MST.
    • Global Min Edge: Absolute lightest edge in Always IN every MST.
    • Threshold Inclusion: Edge is in every MST weight() max weight on minimax path between its endpoints.
    • Multiple MSTs: Different trees, but identical sorted sequence of edge weights.

    Formula Matrix

    Formula Matrix

    MST Edge Count
    Exactly
    Weight Invariance
    invariant under tie-breaking
    Component Counting
    where is multigraph at weight
    Matrix Tree Theorem
    for any
    Cayley's Formula

    If You See This, Do This

    If You See This, Do This

    "lexicographical tie-breaking"
    Sort by vertex label before cycle check
    "starting vertex specified"
    Prim's; update frontier deterministically
    "number of distinct MSTs"
    Isolate components at weight ; count multigraph trees
    "forced edge in MST"
    Contract edge; run MST on remainder; add forced weight
    "all edges weight "
    Count spanning trees of (Matrix Tree / Cayley)
    "swapping edges"
    Check if they form a cycle with same max weight

    Traps and Invariants

    Traps and Invariants

    Global multiplication of equal-weight edges. Check: Isolate independent components at current weight.
    Unique weight unique edge set. Check: Cycle with 2, 2, 3 has weight 4 but two MSTs.
    Equal-weight edge forming cycle with lower weights. Check: Heaviest edge in any cycle cannot be in MST.

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